Carl Pomerance

Carl Pomerance
Carl Pomerance
Phone:
(603) 646-2635
Dept. Fax:
(603) 646-1312
Office:
344 Kemeny Hall
Email:
carl.pomerance@dartmouth.edu
US Mail:
Department of Mathematics
Dartmouth College
Hanover, NH 03755-3551
(603) 646-2415
Brief CV
Books
Lecture Notes on Primality Testing and Factoring:
A Short Course at Kent State University
, C. Pomerance,
MAA Notes 4, Washington, DC, 1984.
Advances in Cryptology: Crypto '87
C. Pomerance, ed., Lecture Notes in Computer Science
293, Springer–Verlag, Berlin, 1988.
Cryptology and computational number theory
C. Pomerance, ed.
Proc. Symp. Appl. Math. 42,
Amer. Math. Soc., Providence, 1990.
Prime numbers: a computational perspective
R. E. Crandall and C. Pomerance,
Springer–Verlag, New York, 2001.
Prime Numbers: a computational perspective, second edition
R. E. Crandall and C. Pomerance, Springer, New York, 2005.
Errata.
Topics in combinatorial number theory: Proceedings of the INTEGERS
Conference 2003 in honor of Tom Brown
B. Landman, M. Nathanson, J. Nesetril, and C. Pomerance, eds.,
DIMATIA, Prague, 2005.
Combinatorial Number Theory: Proceedings of the Integers Conference
2005 in Celebration of the 70th Birthday of Ron Graham
B. Landman, M. Nathanson,
J. Nesetril, R. Nowakowski, and C. Pomerance, eds., De Gruyter, Berlin, 2007.
Analytic number theory (in honor of Helmut Maier's 60th birthday)
C. Pomerance and M. Rassias, eds., Springer, Cham, Switzerland, 2015.
Analytic number theory (in honor of Helmut Maier's 70th birthday)
J. Friedlander, C. Pomerance, and M. Rassias, eds., Springer, to appear.
Some Talks
Covering talk
Talk on covering congruences (Joint Math Meetings, San Diego, January 2008).
Undergrad covering talk
More elementary talk on covering congruences (Spuyten Duyvil Undergraduate
Mathematics Meeting, New York City, April, 2008; Ohio and Michigan
MAA Sections, Spring 2008).
Talk on Euler's function
Talk at University of Georgia, February 2008 and Trinity University, March 2008.
Elementary primality talk
Lucas Lecture at Fibonacci Association Meeting in Patras, Greece, July 2008.
Fields talk
Counting Fields. At Canadian Number Theory Association Meeting
in Waterloo, Canada, July 2008.
Version
for Berkeley Number Theory Seminar, February 2010.
Multiplicative order talk
The multiplicative order mod
, on average.
At the Quebec/Maine number theory conference at Laval University,
Quebec, Canada, October, 2008.
Order and chaos
At the PANTS meeting (yes, a "long PANTS talk"), University of
South Carolina, Columbia, SC, December, 2008.
Version of March, 2009
Brigham Young University.
Version of April, 2009
University of Rochester.
Sociable numbers: new developments on
an ancient problem
Session on the Beauty and Power of Number Theory, Joint Mathematics
Meetings, Washington, DC, January, 2009.
Long version of March, 2009
Brigham Young University undergraduate colloquium.
Long version of April, 2009
Lorentz Center, Leiden, The Netherlands.
The first dynamical system?
AMS Special Session, Boston College, April, 2013.
A 1935 Erdős paper on prime numbers
and Euler's function
at the AMS Central Section meeting, University of Illinois, Urbana, IL,
March, 2009.
Version of July, 2009
, at the
University of Montreal and at the 41st Conference on Combinatorics,
Graph Theory and Computing,
Version of December, 2009
at the West Coast Number Theory Conference.
Discrete Logarithms
Dartmouth Number Theory Seminar, November 19, 2009.
Fixed points for discrete logarithms
41st Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL,
March 2010.
Related talk:
The Pólya–Vinogradov
inequality
Illinois Number Theory Conference in Honor of Harold Diamond, May 21, 2010.
Related talk:
Fixed points for discrete logarithms
ANTS IX, Nancy, France, July 19–23, 2010.
Fibonacci integers
Banff conference in honor of Cam Stewart, May 31, 2010 to June 4, 2010.
Counting in number theory, the Rademacher Lectures, U Penn, September, 2010:
Elementary number theory
Finite cyclic groups
Fibonacci integers
Counting fields
Two problems in combinatorial number theory
Number theory and its applications, Debrecen, Hungary, Oct. 4–8, 2010.
Elliptic curves: applications and problems
IMA Abel Conference, Minneapolis, MN, January, 2011.
Order and chaos
MSRI Arithmetic Statistics Introductory Workshop,
Berkeley, CA, January/February, 2011.
Undergrad version:
Order and chaos
Hudson River Undergraduate Mathematics Conference,
Skidmore College, Saratoga Springs, NY, April 16, 2011.
Grad version:
Order and chaos
Quebec Student Conference, University of Montreal,
May 20, 2011.
Short version:
A problem of Arnold on the average multiplicative order
Maine/Quebec Number Theory Conference, University of Maine,
October 1,2, 2011.
Version of November, 2012
Version of December, 2013
Product-free sets of integers
Integers Conference, Carrollton, GA, October, 2011.
Sums and products
Mathematics & Statistics Colloquium, U. Vermont, December 2, 2011.
Boston Meetings version
Joint Mathematics Meetings, Boston, MA, 2012.
UC Irvine colloquium, February 9, 2012.
Sums and products
International Number Theory Conference in memory of
Alf van der Poorten, AM, March 15, 2012.
Sums and products
Dartmouth Mathematics Society, May 16, 2012.
Sums and products
U. Georgia VIGRE seminar, November 28, 2012.
Sums and products
West Chester U., April 24, 2013, SUNY Albany, April 25, 2013.
Sums and products
Arizona State U., February 27, 2014.
Sums and products
Providence College, April 2, 2014.
What we still don't know about addition
and multiplication
, Johns Hopkins Center for Talented Youth,
May 4, 2013.
What we still don't know about addition
and multiplication
, Richard and Louise Guy Lecture, University
of Calgary, September 12, 2013.
What we still don't know about addition
and multiplication
, Woods Lecture Series, Butler University,
December 3, 2013.
What we still don't know about addition
and multiplication
, Fuzzy Vance Lecture, Oberlin College,
March 20, 2014.
What we still don't know about addition
and multiplication
, Christie Lecture, Bowdoin College,
April 7, 2014.
What we still don't know about addition
and multiplication
, Bay Area Mathematical Adventures, Santa Clara University,
January 23, 2015.
Sets of monotonicity for Euler's function
West Coast Number Theory Conference, Asilomar Conference Center, Pacific Grove, CA, December, 2011.
The range of Carmichael's function
AMS Sectional Meeting, U. Hawaii, Honolulu, March, 2012.
Balanced subgroups of the multiplicative group
Dartmouth Number Theory Seminar, April 26, 2012.
Balanced subgroups of the multiplicative group
CNTA, June, 2012.
Balanced subgroups of the multiplicative group
Quebec/Maine Number Theory Conference, September, 2012.
Balanced subgroups of the multiplicative group
Central Section AMS Meeting, U. Akron, Akron, OH, October, 2012.
Balanced subgroups of the multiplicative group
Palmetto Number Theory Symposium, December 2, 2012.
Some statistical problems concerning the arithmetic
functions σ and ϕ,
Joint Mathematics Meetings, Special Session on Arithmetic Statistics,
San Diego, CA, January, 2013.
Erdős, van der Corput, and the birth
of covering congruences,
Joint Mathematics Meetings, Special Session on Covering Congruences,
San Diego, CA, January, 2013.
Paul Erdős and the rise of
statistical thinking in elementary number theory,
Erdős Centennial Conference, Budapest, July 1–5, 2013.
Paul Erdős and the rise of statistical
thinking in elementary number theory,
Version of 1/15/14 at the Joint Math Meetings, Baltimore.
The set of values of an arithmetic function,
Mathematical Congress of the Americas, Guanajuato, Mexico, August 9, 2013.
The set of values of an arithmetic function,
Integers Conference (marking the Erdős centennial), University of West Georgia,
Carrollton, GA, October 23–27, 2013.
The ranges of various familiar functions,
CNTA XIII, Carleton University, Ottawa, Canada, June 20, 2014.
The ranges of some familar arithmetic functions,
Maine/Quebec Number Theory Conference, Orono, ME, October 3-4, 2015.
Square values of Euler's function,
SCHOLAR conference in honour of M. Ram Murty, University of Montreal, October
15–17, 2013.
Amicable numbers,
Illinois Conference in Memory of Felice and Paul Bateman and
Heini Halberstam, June 5-7, 2014.
Amicable numbers,
Brown University Algebra Seminar, April 6, 2015.
The statistics of elementary number theory,
2014 NCTS Conference on the Impact of Computation in Number Theory,
30 July, 2014 to 3 August, 2014, National Tsing Hua University, Hsinchu, Taiwan.
Statistics in elementary number theory,
CRM Workshop on "Statistics and elementary number theory", Montreal, 15-19 September, 2014.
The first function
U. Georgia colloquium, December 3, 2014.
The first function
Middlebury College Seminar, April 21, 2015.
The first function
Connections in Discrete Mathematics, A celebration of the work of Ron Graham,
Simon Fraser U., June 15 - 19, 2015
The sum-of-proper-divisors function
BC–MIT Number Theory Seminar, September 15, 2015.
The first function
Cal State Chico Colloquium, February 5, 2016.
The first function
Dartmouth Colloquium, May 26, 2016.
The first function
Western Michigan U., Colloquium, October 14, 2016.
The CRM Aisenstadt Chair Lectures, December 8–12, 2014, Montréal.
The ranges of some familiar functions
December 8, 2014.
The first function
December 11, 2014.
Amicable numbers
December 12, 2014
Letters from the master: my correspondence
with Paul Erdős
Lecture at History of Mathematics
Special Interest Group of the Mathematical Association of America,
Joint Mathematics Meetings, San Antonio, TX, January 10, 2015.
Here
are some sample letters that are
discussed near the end of the talk.
Random number theory
Lecture at Random Roads: A celebration of Joel Spencer's 70th birthday,
NYU, April 30, 2016.
'Rithmetic revisited: what we still don't know about + and ×
Math Encounters, National Museum of Mathematics, June 1, 2016.
The first function and the Guy-Selfridge conjecture
CNTA Calgary, Guy Session, June 21, 2016.
The first dynamical system (with a short feature)
Summer school on fractal geometry and complex dimensions, Cal Poly San Luis Obispo, June 27, 2016.
Why the ABC conjecture
Kummer classes and anabelian geometry, U. Vermont, September 10-11, 2016.
Euclidean prime generators
Dartmouth Number Theory Seminar, October 4, 2016 and Integers Conference,
U. West Georgia, October 6, 2016.
Euclidean prime generators
West Coast Number Theory Conference, Pacific Grove, CA, December 2016.
What we still don't know about addition and
multiplication
Undergraduate Lecture Series, Michigan State U., October 11, 2016.
The ranges of some familiar arithmetic functions
Michigan State U. Colloquium, October 13, 2016.
The ranges of some familiar arithmetic functions
Max Planck Institute for Mathematics, November 2, 2016.
The ranges of some familiar arithmetic functions
, MSRI, May 7, 2017.
The first dynamical system
Charles U. (Prague) Seminar, November 8, 2016.
Random number theory
Charles U. (Prague) Colloquium, November 8, 2016.
U. Georgia talks, March 22-28, 2017
Random number theory
Euclidean prime generators
What we still don't know about addition and multiplication
The first dynamical system
New results on an ancient function
Maine/Quebec Number Theory Conference, October 14,15, 2017.
New results on an ancient function
Joint Math Meetings 2018, San Diego, Special Session on Computational
Methods in Combinatorics and Number Theory.
The aliquot constant
West Coast Number Theory Conference, December 16-20, 2017.
What we still don't know about addition and multiplication
Leonard C. Sulski Memorial Lecture and MAA Northeastern Section Dinner Meeting,
College of the Holy Cross, Worcester, MA, April 23, 2018.
The Erdős problem on primitive sets
CRM workshop on Probability in Number Theory, U. de Montreál,
May 28, 2018. (See draft of paper on this topic with some updated results.)
The Erdős problem on primitive sets
Integer Conference, Augusta, GA, October 3, 2018.
The Erdős problem on primitive sets
West Coast Number Theory Conference, Chico, CA, December 15-19, 2018.
The Erdős problem on primitive sets
Joint Mathematics Meetings, Baltimore, MD, January 15-19, 2019.
Random number theory
CMS Summer Meeting, U. New Brunswick, June 4, 2018.
What we still don't know about addition and multiplication
Evans–Bourdon Lecture, Washington & Lee U., October 9, 2018.
Primality testing: then and now
Celebrating 75 years of Mathematics of Computation, ICERM, November 1–3, 2018, Providence.
What we still don't know about addition
and multiplication
Trjitzinsky Lecture 1, U. Illinois Urbana-Champaign,
November 27, 2018.
Random number theory
Trjitzinsky Lecture 2, U. Illinois Urbana-Champaign,
November 28, 2018.
Primality testing: then and now
Trjitzinsky Lecture 3, U. Illinois Urbana-Champaign,
November 29, 2018.
Primality testing: then and now
Boise State University, February 20, 2019.
Erdős and primitive sets
Erdős Lecture Series, University of Memphis, September 12–15, 2019.
Cyclotomic polynomials: problems and results
PaNTS XXXIII, Clemson, SC, December 14–15, 2019.
Glasby's cyclotomic ordering conjecture
West Coast Number Theory Conference, Asilomar, December 16–20, 2019.
The first function
Bay Area Mathematical Adventures, Santa Clara U., January 31, 2020.
Is 73 the best number?
MAA Golden Section Meeting, Mills College, February 29, 2020.
Symmetric primes
CANT, New York City (via Zoom), June 3, 2020.
Practical numbers
Number Theory Web Seminar (via Zoom), August 13, 2020.
Aliquot sequences
Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy,
University of Calgary (via Zoom), October 2, 2020.
Denominators of Bernoulli numbers
Dartmouth Number Theory Seminar (via Zoom), May 25, 2021.
Is 73 the best number?
MAA Northeastern Section Meeting (via Zoom), June 5, 2021.
Coprime matchings and permutations
Santa Clara Math and CS Department, April 12, 2022.
Coprime permutations
Dartmouth College Combinatorics Seminar, May 17, 2022.
Permutations and
arithmetic
Number Theory Conference Debrecen, July 4 to 8, 2022.
Permutations and
arithmetic
A celebration of analytic number theory, a conference in honor of Andrew Granville, September 6, 2022.
The Sieve of Eratosthenes and Rough Numbers
West Coast Number Theory Conference, December 16, 2022.
Digits
Athens/Atlanta Number Theory Seminar, February 6, 2023, Athens, GA.
Digits
Arithmetic, Algebra, and Algorithms, ICMS, April 10–14, 2023, Edinburgh, Scotland.
What we still don't know about addition
and multiplication
Ross Program, July 18, 2023, Indiana and Ohio.
The shifted-prime divisor function
PaNTS XXXVI (in memory of Kevin James), Clemson, October 21–22, 2023.
Matchable numbers
CANT, CUNY, May 2024.
Cyclotomic primes
Dartmouth Algebra/Number Theory Seminar, May 2025.
Is 73 the best number?
Dartmouth Mathematics Society, May 2025.
Is number theory a science?
New York Number Theory Seminar, December, 2025.
The Erdős–Straus conjecture
Number Theory Web Seminar, March 12, 2025.
The Erdős–Straus conjecture
Alladi 70 Conference, University of Florida, March 18-22,2026
Papers
Odd perfect numbers are divisible by at least seven distinct primes
C. Pomerance,
Acta Arith.
25
(1974),
265–300.
On Carmichael's conjecture
C. Pomerance,
Proc. Amer. Math. Soc.
43
(1974),
297–298.
A search for elliptic curves with large rank
D.E. Penney and C. Pomerance,
Math. Comp.
28
(1974),
851–853.
714 and 715
C. Nelson, D.E. Penney, and C. Pomerance,
J. Rec. Math.
(1974),
87–89.
Three elliptic curves with rank at least seven
D.E. Penney and C. Pomerance,
Math. Comp.
29
(1975),
965–967.
The second largest prime factor of an odd perfect number
C. Pomerance,
Math. Comp.
29
(1975),
914–921.
On the congruences
σ(
) ≡
(mod
and
(mod ϕ(
))
C. Pomerance,
Acta Arith.
26
(1975),
265–272.
On an interesting property of 112359550561797752809
J.L. Hunsucker and C. Pomerance,
Fibonacci Quarterly
13
(1975),
331–333.
There are no odd super perfect numbers less than 7 x 10
24
J.L. Hunsucker and C. Pomerance,
Indian J. Math.
17
(1975),
107–120.
Some new results on odd perfect numbers
G.G. Dandapat, J.L. Hunsucker, and C. Pomerance,
Pacific J. Math.
57
(1975),
359–364.
On multiply perfect numbers with a special property
C. Pomerance,
Pacific J. Math.
57
(1975),
511–517.
On composite
for which ϕ(
)|
–1, I
C. Pomerance,
Acta Arith.
28
(1976),
387–389.
Multiply perfect numbers, Mersenne primes and effective computability
C. Pomerance,
Math. Ann.
226
(1977),
195–206.
On a tiling problem of R. B. Eggleton
C. Pomerance,
Discrete Math.
18
(1977),
63–70.
On composite
for which ϕ(
)|
–1, II
C. Pomerance,
Pacific J. Math.
69
(1977),
177–186.
On the distribution of amicable numbers
C. Pomerance,
J. reine angew. Math.
293/294
(1977),
217–222.
On the largest prime factors of
and
+1
P. Erdős and C. Pomerance,
Aequationes Math.
17
(1978),
311–321.
On a class of relatively prime sequences
P. Erdős, D.E. Penney, and C. Pomerance,
J. Number Theory
10
(1978),
451–474.
The prime number graph
C. Pomerance,
Math. Comp.
33
(1979),
399–408.
On a problem of Evelyn–Linfoot and Page in additive number theory
C. Pomerance and D. Suryanarayana,
Publ. Math. Debrecen
26
(1979),
237–244.
Nearly parallel vectors
H.G. Diamond and C. Pomerance,
Mathematika
26
(1979),
258–268.
Some number theoretic matching problems
C. Pomerance,
Proceedings of the Queen's Number Theory Conference
P. Ribenboim, ed.,
Queen's Papers in Pure and Applied Mathematics, No. 54,
Kingston, Canada, 1979,
237–247.
Collinear subsets of lattice point sequences — an analogue of
Szemerédi's theorem,
C. Pomerance,
J. Combinatorial Theory (A)
28
(1980),
140–149.
A note on the least prime in an arithmetic progression
C. Pomerance,
J. Number Theory
12
(1980),
218–223.
The pseudoprimes to
25 x 10
C. Pomerance, J.L. Selfridge, and S.S. Wagstaff, Jr.,
Math. Comp.
35
(1980),
1003–1026.
Matching the natural numbers up to
with distinct multiples
in another interval
P. Erdős and C. Pomerance,
Nederl. Akad. Wetensch. Proc. Ser. A
83
(1980),
147–161.
Proof of D.J. Newman's coprime mapping conjecture
C. Pomerance and J.L. Selfridge,
Mathematika
27
(1980),
69–83.
Popular values of Euler's function
C. Pomerance,
Mathematika
27
(1980),
84–89.
Sets on which an entire function is determined by its range
H.G. Diamond, C. Pomerance, and L. Rubel,
Math. Z.
176
(1981),
383–398.
On the distribution of amicable numbers, II
C. Pomerance,
J. reine angew. Math.
325
(1981),
183–188.
The arithmetic mean of the divisors of an integer
P.T. Bateman, P. Erdős, C. Pomerance, and E.G. Straus,
Analytic Number Theory Proceedings, Philadelphia 1980
M. I. Knopp, ed.,
Lecture Notes in Math.
899
(1981),
197–220.
On the distribution of pseudoprimes
C. Pomerance,
Math. Comp.
37
(1981),
587–593.
Recent developments in primality testing
C. Pomerance,
Math. Intelligencer
(1981),
97–105.
A new lower bound for the pseudoprime counting function
C. Pomerance,
Illinois J. Math.
26
(1982),
4–9.
The search for prime numbers
C. Pomerance,
Scientific American
247
No. 6 (1982),
136–144.
Analysis and comparison of some integer factoring algorithms
C. Pomerance,
Computational Methods in Number Theory, Part I
H.W. Lenstra, Jr. and R. Tijdeman, eds.,
Math. Centre Tract
154,
Amsterdam, 1982,
89–139.
On distinguishing prime numbers from composite numbers
L.M. Adleman, C. Pomerance, and R.S. Rumely,
Annals Math.
117
(1983),
173–206.
An analogue of Grimm's problem of finding distinct prime factors
of consecutive integers
P. Erdős and C. Pomerance,
Utilitas Math.
24
(1983),
45–65.
On a problem of Oppenheim concerning `Factorisatio Numerorum'
E.R. Canfield, P. Erdős, and C. Pomerance,
J. Number Theory
17
(1983),
1–28.
Implementation of the continued fraction integer factoring algorithm
C. Pomerance and S.S. Wagstaff, Jr.,
Congressus Numerantium
37
(1983),
99–117.
On the longest simple path in the divisor graph
C. Pomerance,
Proc. Southeastern Conf. Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983
Cong. Num.
40
(1983),
291–304.
Moduli
for which there are many small primes congruent to
modulo
P.T. Bateman and C. Pomerance,
Publ. Math. d'Orsay
83.04
(1983),
8–19.
Lecture notes on primality testing and factoring — A short
course at Kent State University
C. Pomerance, (based on notes by S. M. Gagola, Jr.),
MAA Notes
(1984).
Are there counter-examples to the Baillie—PSW primality test
in
DOPO LE PAROLE aangeboden aan DR. A. K. LENSTRA
, H. W. Lenstra, jr, J. K. Lenstra, and
P. van Emde Boas, eds., Amsterdam, 1984. (Re-typeset by Jon Grantham.)
New ideas for factoring large integers
C. Pomerance, J. W. Smith, and S. S. Wagstaff, Jr.,
Advances in Cryptology, Proc. Crypto 83
D. Chaum, ed.,
Plenum Press, New York,
1984,
81–85.
Estimates for certain sums involving the largest prime factor of
an integer
A. Ivic and C. Pomerance,
Proc. Colloquium on Number Theory
34
(1981),
Topics in Classical Number Theory,
North Holland,
1984,
769–789.
On the size of the coefficients of the cyclotomic polynomial
P. T. Bateman, C. Pomerance, and R. C. Vaughan,
Proc. Colloquium on Number Theory
34
(1981),
Topics in Classical Number Theory,
North Holland,
1984,
171–202.
View obstruction problems, III
T. W. Cusick and C. Pomerance,
J. Number Theory
19
(1984), 131–139.
The normal number of prime factors of ϕ(
P. Erdős and C. Pomerance,
Rocky Mtn. J. Math.
15
(1985),
343–352.
On locally repeated values of certain arithmetic functions, I
P. Erdős, C. Pomerance, and A. Sárközy,
J. Number Theory
21
(1985),
319–332.
Multiplicative relations for sums of initial
-th powers
D. E. Penney and C. Pomerance,
Amer. Math. Monthly
92
(1985),
729–731.
On the distribution of round numbers
C. Pomerance,
Number Theory Proceedings, Ootacamund, India 1984
K. Alladi, ed.,
Lecture Notes in Math.
1122
(1985),
173–200.
The quadratic sieve factoring algorithm
C. Pomerance,
Advances in Cryptology, Proceedings of Eurocrypt 84, Paris, 1984
T. Beth. N. Cot, and I. Ingemarsson, eds.,
Lecture Notes in Computer Sci.
209
(1985),
169–182.
On the Schnirelmann and asymptotic densities of
certain sets of non-mulitples
P. Erdős, C. B. Lacampagne, C. Pomerance, and J. L. Selfridge,
Proceedings of the Southeast Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1985
Congressus Numerantium
48
(1985),
67–79.
On sums involving reciprocals of the largest prime factor of an integer
P. Erdős, A. Ivic, and C. Pomerance,
Glasnik Math.
21
(1986),
283–300.
On the number of false witnesses for a composite number
P. Erdős and C. Pomerance,
Math. Comp.
46
(1986),
259–279.
On primitive divisors of Mersenne numbers
C. Pomerance
Acta Arith.
46
(1986),
355–367.
On the distribution of the values of Euler's function
C. Pomerance,
Acta Arith.
47
(1986),
63–70.
On locally repeated values of certain arithmetic functions, II
P. Erdős, C. Pomerance, and A. Sárközy,
Acta Math. Hungarica
49
(1987),
251–259.
On the average number of groups of square-free order
C. Pomerance,
Proc. Amer. Math. Soc.
99
(1987),
223–231.
The smallest n-uniform hypergraph with positive discrepancy
N. Alon, D. J. Kleitman, C. Pomerance, M. Saks, and P. Seymour,
Combinatorica
(1987),
151–160.
On locally repeated values of certain arithmetic functions, III
P. Erdős, C. Pomerance, and A. Sárközy,
Proc. Amer. Math. Soc.
101
(1987),
1–7.
Very short primality proofs
C. Pomerance,
Math. Comp.
48
(1987),
315–322.
Fast, rigorous factorization and discrete logarithm algorithms
C. Pomerance,
Discrete algorithms and complexity
D. S. Johnson, T. Nishizeki, A. Nozaki, H. S. Wilf, eds.,
Academic Press,
Orlando, Florida, 1987,
pp. 119–143.
On products of sequences of integers
C. Pomerance and A. Sárközy,
Coll. Math. Soc. Janos Bolyai
51
(1987),
447–463.
A pipeline architecture for factoring large integers with the
quadratic sieve algorithm
C. Pomerance, J. W. Smith, and R. Tuler,
SIAM J. Comput.
17
(1988),
387–403.
On homogeneous multiplicative hybrid problems in number theory
C. Pomerance and A. Sárközy,
Acta Arith.
49
(1988),
291–302.
On the number of distinct values of Euler's ϕ-function
H. Maier and C. Pomerance,
Acta Arith.
49
(1988),
263–275.
On divisors of sums of integers, III
C. Pomerance, A. Sárközy, and C. L. Stewart,
Pacific J. Math.
133
(1988),
363–379.
The generation of random numbers that are probably prime
P. Beauchemin, G. Brassard, C. Crépeau, C. Goutier, and C. Pomerance,
Journal of Cryptology
1 (1988),
53–64.
Two methods in elementary analytic number theory
C. Pomerance,
Number theory and applications
R. A. Mollin, ed.,
Kluwer Academic Publishers,
Dordrecht, 1989,
pp. 135–161.
On the composition of the arithmetic functions σ and ϕ
C. Pomerance,
Colloq. Math.
58 (1989),
11–15.
The probability that a random probable prime is composite
S.H. Kim and C. Pomerance,
Math. Comp.
53 (1989),
721–741.
Fonction zêta de Riemann et conjecture de Weyl–Berry pour les
tambours fractals
M. L. Lapidus and C. Pomerance,
C. R. Acad. Sci. Paris (Ser. I)
310 (1990),
343–348.
On the normal behavior of the iterates of some arithmetic functions
P. Erdős, A. Granville, C. Pomerance, and C. Spiro,
Analytic Number Theory, Proc. Conf. in honor of Paul T. Bateman
B. C. Berndt, et al. eds.,
Birkhauser,
Boston, 1990,
pp. 165–204.
Unusually large gaps between consecutive primes
H. Maier and C. Pomerance,
Trans. Amer. Math. Soc.
322 (1990),
201–237.
On the least prime in certain arithmetic progressions
A. Granville and C. Pomerance,
J. London Math. Soc. (2)
41 (1990),
193–200.
Factoring
C. Pomerance,
Cryptology and Computational Number Theory
C. Pomerance, ed.,
Proc. Symp. Appl. Math.
42,
Amer. Math. Soc.
Providence, 1990.
Cryptology and computational number theory — an introduction
C. Pomerance,
Cryptology and Computational Number Theory
C. Pomerance, ed.,
Proc. Symp. Appl. Math.
42,
Amer. Math. Soc., Providence, 1990.
On a theorem of Besicovitch: values of arithmetic functions
that divide their arguments
P. Erdős and C. Pomerance,
Indian J. Math.
32 (1990),
279–287.
On prime divisors of Mersenne numbers
P. Erdős, P. Kiss, and C. Pomerance,
Acta Arith.
57 (1991),
267–281.
Carmichael's lambda function
P. Erdős, C. Pomerance, and E. Schmutz,
Acta Arith.
58 (1991),
363–385.
The distribution of Lucas and elliptic pseudoprimes
D.M. Gordon and C. Pomerance,
Math. Comp.
57 (1991),
825–838.
Grandes déviations pour certaines fonctions arithmétiques
M. Balazard, J.L. Nicolas, C. Pomerance, and G. Tenenbaum,
J. Number Theory
40 (1992),
146–164.
The distribution of smooth numbers in arithmetic progressions
A. Balog and C. Pomerance,
Proc. Amer. Math. Soc.
115 (1992),
33–43.
A rigorous time bound for factoring integers
H. W. Lenstra, Jr. and C. Pomerance,
J. Amer. Math. Soc.
5 (1992),
483–516.
Reduction of huge, sparse matrices over a finite field via
created catastrophes
C. Pomerance and J. W. Smith,
Experimental Math.
1 (1992),
90–94.
The Riemann zeta function and the one dimensional Weyl-Berry
conjecture for fractal drums
M. L. Lapidus and C. Pomerance,
Proc. London Math. Soc. (3)
66 (1993),
41–69.
Average case error estimates for the strong probable prime test
I. Damgard, P. Landrock, and C. Pomerance,
Math. Comp.
61 (1993),
177–194.
Carmichael numbers
C. Pomerance,
Nieuw Arch. Wisk.
11 (1993),
199–209.
On elements of sumsets with many prime factors
P. Erdős, C. Pomerance, A. Sárközy, and C. L. Stewart,
J. Number Theory
44 (1993),
93–104.
An upper bound in Goldbach's conjecture
J.M. Deshouillers, A. Granville, W. Narkiewicz, and C. Pomerance,
Math. Comp.
61 (1993),
209–213.
Factoring integers with the number field sieve
J. Buhler, H. W. Lenstra, Jr., and C. Pomerance,
The development of the number field sieve
A. K. Lenstra and H. W. Lenstra, Jr., eds.,
Lecture Notes in Math.
1554,
pp. 50–94,
Springer-Verlag, Berlin, 1993.
A hyperelliptic smoothness test. I
H. W. Lenstra, Jr., J. Pila, and C. Pomerance,
Phil. Trans. R. Soc. London A
345 (1993),
397–408.
Sixes and sevens
C. Pomerance,
Missouri J. Math. Sci.
6 (1994),
62–63.
There are infinitely many Carmichael numbers
W. R. Alford, A. Granville, and C. Pomerance,
Ann. of Math.
(2)
139 (1994),
703–722.
On the difficulty of finding reliable witnesses
W. R. Alford, A. Granville, and C. Pomerance,
Algorithmic Number Theory Proceedings (ANTS-I)
L. M. Adleman and M.-D. Huang, eds.,
Lecture Notes in Computer Sci.
877 (1994),
Springer-Verlag, Berlin,
pp. 1–16.
Dickson polynomials with few fixed points in a finite field
C. Pomerance,
J. Sichuan U. (Natural Science Ed.)
31 (1994),
460–464.
On a conjecture of R. L. Graham
F. Y. Cheng and C. Pomerance,
Rocky Mtn. J. Math.
24 (1994),
961–975.
The number field sieve
C. Pomerance,
Mathematics of Computation, 1943–1993, Fifty Years of Computational Mathematics
W. Gautschi, ed.,
Proc. Symp. Appl. Math.
48,
American Mathematical Society,
Providence, 1994,
pp. 465–480.
Counting the integers factorable via cyclotomic methods
C. Pomerance and J. Sorenson,
J. Algorithms
19 (1995),
250–265.
On a conjecture of Crandall concerning the
qx
+1 problem
Z. Franco and C. Pomerance,
Math. Comp.
64 (1995),
1333–1336.
Implementing the self initializing quadratic sieve on
a distributed network
W.R. Alford and C. Pomerance,
Number Theoretic and Algebraic Methods in Computer Science, Proc. of Int'l Moscow Conference, June-July, 1993
A. J. van der Poorten, I. Shparlinski, H. G. Zimmer, eds.,
World Scientific,
1995,
pp. 163–174.
Combinatorial number theory
C. Pomerance and A. Sárközy,
Handbook of Combinatorics
R. L. Graham, M. Grötschel, L. Lovász, eds.,
Elsevier Science B.V., 1995,
pp. 967–1018.
On the role of smooth numbers in number theoretic algorithms
C. Pomerance,
Proceedings of the Intenational Congress of Mathematicians, Zurich, Switzerland 1994
Birkhauser Verlag,
Basel, 1995,
pp. 411–422.
Counterexamples to the modified Weyl-Berry conjecture
M.L. Lapidus and C. Pomerance,
Math. Trans. Cambridge Phil. Soc.
119 (1996),
167–178.
Symmetric and asymmetric primes
P. Fletcher, W. Lindgren, and C. Pomerance,
J. Number Theory
58 (1996),
89–99.
Multiplicative independence for random integers
C. Pomerance,
Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, Vol. 2
B. Berndt, H. Diamond, A. Hildebrand, eds.,
Birkhauser, Boston, 1996,
pp. 703–711.
On the divisors of
P. Erdős, S.W. Graham, A. Ivic, and C. Pomerance,
Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, Vol. 1
B. Berndt, H. Diamond, A. Hildebrand, eds.,
Birkhauser, Boston, 1996,
pp. 337–355.
A tale of two sieves
C. Pomerance,
The Notices of the Amer. Math. Soc.
43 (1996),
1473–1485.
On primes recognizable in deterministic polynomial time
S. V. Konyagin and C. Pomerance,
The mathematics of Paul Erdős
R. L. Graham and J. Nesetril, eds.,
Springer-Verlag, Berlin, 1997,
pp. 176–198.
See #188 for an update article.
A search for Wieferich and Wilson primes
R. Crandall, K. Dilcher, and C. Pomerance,
Math. Comp.
66 (1997),
433–449.
On locally repeated values of certain arithmetic functions, IV
P. Erdős, C. Pomerance, and A. Sárközy,
The Ramanujan J.
1 (1997),
227–241.
Automaticity II: Descriptional complexity in the unary case
C. Pomerance, J.M. Robson, and J. Shallit,
Theoretical Computer Sci.
180 (1997),
181–201.
Paul Erdős, number theorist extraordinaire
C. Pomerance,
The Notices of the Amer. Math. Soc.
45 (1998),
19–23.
Rigorous discrete logarithm computations in finite fields via
smooth polynomials
R. Lovorn Bender and C. Pomerance,
AMS/IP Studies in Advanced Mathematics
7 (1998),
221–232.
Euler's function in residue classes
T. Dence and C. Pomerance,
The Ramanujan Journal
2 (1998),
7–20.
On the distribution of champs
A. Ivic and C. Pomerance,
Proceedings of the Fifth Conference of the Canadian Number Theory
Association
R. Gupta and K.S. Williams, eds.,
CRM Proc.
19 (1999),
133–139.
Residue classes free of values of Euler's function
K. Ford, S. V. Konyagin, and C. Pomerance,
Number Theory in Progress
K. Gyory, H. Iwaniec, and J. Urbanowicz, eds.,
vol. 2, de Gruyter, Berlin and New York, 1999,
pp. 805–812.
On the solutions to ϕ(
) = ϕ(
S.W. Graham, J.J. Holt, and C. Pomerance,
Number Theory in Progress
K. Gyory, H. Iwaniec, and J. Urbanowicz, eds.,
vol. 2, de Gruyter, Berlin and New York, 1999,
pp. 867–882.
Primes and factorization
J. Grantham and C. Pomerance,
Handbook of Discrete Mathematics
K.H. Rosen, ed.,
CRC Press, 1999.
Small values of the Carmichael function and cryptographic applications
J. Friedlander, C. Pomerance, and I. E. Shparlinski,
Proc. Workshop on Cryptography and Computational Number Theory (CCNT'99)
K.-Y. Lam, I. E. Shparlinski, H. Wang, and C. Xing, eds.,
Birkhäuser, 2001,
pp. 25–32.
The expected number of random elements to generate a finite
abelian group
C. Pomerance,
Periodica Mathematica Hungarica
43 (2001),
191–198.
Period of the power generator and small values of the Carmichael
function
J. Friedlander, C. Pomerance, and I. E. Shparlinski,
Math. Comp.
70 (2001), 1591–1605.
Corrigendum
op. cit.
, 71 (2002), 1803–1806.
Two contradictory conjectures concerning Carmichael
numbers
, A. Granville and C. Pomerance,
Math. Comp.
71 (2001), 883–908.
On the problem of uniqueness for the maximal Stirling
number(s) of the second kind
, E.R. Canfield and C. Pomerance,
Integers
, 2
(2002), paper A1, 13 pp.
(The published form of this paper was somewhat corrupted. The version
here also corrects a small error in Section 4. Posted February, 2013.)
On some problems of Makowski–Schinzel and Erdős
concerning the arithmetical functions ϕ and σ
F. Luca and C. Pomerance,
Colloq. Math.
, 92 (2002), 111–130.
See also
this
Smooth orders and cryptographic applications
C. Pomerance and I.E. Shparlinski,
Proc. ANTS-V, Sydney, Australia
Springer Lecture Notes in Computer Science 2369, (2002), pp. 338–348.
A hyperelliptic smoothness test. II
H. W. Lenstra, Jr., J. Pila, and C. Pomerance,
Proc. London Math. Soc.
(3) 84 (2002), 105–146.
Ruth–Aaron numbers revisited
, C. Pomerance,
Paul Erdős and his Mathematics,
(Budapest, 1999)
, Bolyai Soc. Math. Stud. 11,
János Bolyai Math. Soc., Budapest, 2002, pp. 567–579.
Primitive roots: a survey
, S. Li and C. Pomerance,
in
New Aspects of Analytic Number Theory
(RIMS Kokyuroku No. 1274) (Y. Tanigawa, ed.), and also in
Dev. Math. 8, pp. 219–231,
Kluwer Academic Publishers, Dordrecht 2002.
On generalizing Artin's conjecture on primitive roots to
composite moduli
, S. Li and C. Pomerance,
J. Reine Angew. Math.
556 (2003), 205–224.
Timed fair exchange of arbitrary signatures
, J. A. Garay and C. Pomerance,
in
Financial Cryptography, 7th International Conference, FC 2003
Lecture Notes in Computer Science
2742
, Springer, New York, 2003, pp. 190–207.
Multiplicative structure of values of the Euler function
W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski,
in
High Primes and Misdemeanours: Lectures in Honour of the
Sixtieth Birthday of Hugh Cowie Williams
(A. Van der Poorten, ed.),
Fields Inst. Comm. 41 (2004), pp. 29–47.
Heuristics for class numbers of prime-power real cyclotomic fields
J. Buhler, C. Pomerance, and L. Robertson,
in
High Primes and Misdemeanours: Lectures in Honour of the
Sixtieth Birthday of Hugh Cowie Williams
(A. Van der Poorten, ed.),
Fields Inst. Comm. 41 (2004), pp. 149–157.
Prime numbers and the search for extraterrestrial intelligence
C. Pomerance,
in
Mathematical Adventures for Students and Amateurs
, D. Hayes and T. Shubin, eds.,
M.A.A., 2004, pp. 1–4.
The largest prime factor of a Mersenne number
L. Murata and C. Pomerance, in
Number Theory, CNTA Proceedings, Montreal, 2002
CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004, pp. 209–218.
On the binary expansions of algebraic numbers
D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance,
J. Théorie des Nombres Bordeaux
16 (2004), 487–518.
On the distribution in residue classes of integers with
a fixed sum of digits
, C. Mauduit, C. Pomerance, and A. Sárközy),
Ramanujan J., special issue in honor of J.-L. Nicolas
9 (2005), 45–62.
Products of ratios of consecutive integers
R. de la Bretèche, C. Pomerance, and G. Tenenbaum,
Ramanujan J., special issue in honor of J.-L. Nicolas
9 (2005), 131–138.
The iterated Carmichael λ-function and the number of cycles of the power generator
G. Martin and C. Pomerance,
Acta Arith.
118 (2005), 305–335.
On the period of the linear congruential and power generators
P. Kurlberg and C. Pomerance,
Acta Arith.
119 (2005), 149–169.
Extended abstract
with title "Lower bounds on the period of some pseudorandom number generators".
In Proceedings of Conference on Algorithmic Number Theory 2007, vol. 46 of TUCS
Gen. Pub., pages 74–81. Turku Cent. Comput. Sci., Turku, Finland, 2007.
Finding the group structure of elliptic curves over finite fields
J. B. Friedlander, C. Pomerance, and I. E. Shparlinski,
Bull. Austral. Math. Soc.
72 (2005), 251–263.
On the average number of divisors of the Euler function
F. Luca and C. Pomerance,
Publ. Math. Debrecen
, 70 (2007), 125–148.
Corrigendum
, submitted for publication.
Sieving by large integers and covering systems of congruences
M. Filaseta, K. Ford, S. V. Konyagin, C. Pomerance, and G. Yu,
J. Amer. Math. Soc.
, 20 (2007), 495–517.
Maximal height of divisors of
–1
C. Pomerance and N. C. Ryan,
Illinois J. Math.
, 51 (2007), 597–604.
Irreducible radical extensions and Euler-function chains
F. Luca and C. Pomerance, pp. 351–362 in
Combinatorial Number Theory
Landman et al., eds.,
de Gruyter, 2007, and in
Integers
, 7(2) (2007), paper A25.
Smooth numbers and the quadratic sieve
, C. Pomerance,
in
Algorithmic number theory
, J. P. Buhler
and P. Stevenhagen, eds., Math. Sci. Res. Inst. Pub. 44,
Cambridge U. Press, New York, 2008, pp. 69–81.
Elementary thoughts on discrete logarithms
, C. Pomerance,
in
Algorithmic number theory
, J. P. Buhler
and P. Stevenhagen, eds., Math. Sci. Res. Inst. Pub. 44,
Cambridge U. Press, New York, 2008, pp. 385–396.
Computational number theory
C. Pomerance, in
Princeton Companion to Mathematics
, W. T. Gowers, ed.,
Princeton U. Press, Princeton, New Jersey, 2008, pp. 348–362.
On the proportion of numbers coprime to a given integer
P. Erdős, F. Luca, and C. Pomerance,
Proceedings of the
Anatomy of Integers Conference, Montreal, March 2006
J.-M. De Koninck, A. Granville, F. Luca, eds.,
CRM Proceedings
and Lecture Notes
, vol. 46 (2008), 47–64.
Sets with prescribed arithmetic densities
F. Luca, C. Pomerance, and S. Porubsky,
Uniform Distribution Theory
3 (2008), 67–80.
On pseudosquares and pseudopowers
C. Pomerance and I. E. Shparlinski,
Combinatorial Number Theory, Proceedings
of Integers Conference 2007
, de Gruyter, Berlin, 2009, pp. 171–184.
On the range of the iterated Euler function
F. Luca and C. Pomerance,
Combinatorial Number Theory, Proceedings
of Integers Conference 2007
, de Gruyter, Berlin, 2009, pp. 101–116.
On Giuga numbers
F. Luca, C. Pomerance, and I. E. Shparlinski,
Int. J. Mod. Math.
4 (2009), 13–28.
On the distribution of sociable numbers
M. Kobayashi, P. Pollack, and C. Pomerance,
J. Number Theory
129 (2009), 1990–2009.
On the Artin–Carmichael primitive root problem on
average
, S. Li and C. Pomerance,
Mathematika
55 (2009),
167–176.
On the smallest pseudopower
J. Bourgain, S. V. Konyagin, C. Pomerance, and I. E. Shparlinski,
Acta Arith.
140 (2009), 43–55.
A remark on Giuga's conjecture and Lehmer's totient problem
W. D. Banks, C. W. Nevans, and C. Pomerance,
Albanian J. Math.
3 (2009), 81–85.
On the distribution of pseudopowers
S. V. Konyagin, C. Pomerance, and I. E. Shparlinski,
Canad. J. Math.
62 (2010), 582–594.
Rank statistics for a family of elliptic curves over a function field
C. Pomerance and I. E. Shparlinski,
Pure Appl. Math. Q.
6 (2010), 21–40.
Primality testing: variations on a theme of Lucas
C. Pomerance, in the Proceedings of the 13th Meeting of the
Fibonacci Association,
Congressus Numerantium
201 (2010), 301–312.
Error estimates for the Davenport–Heilbronn theorems
K. Belabas, M. Bhargava, and C. Pomerance,
Duke Math. J.
153 (2010), 173–210.
Common values of the arithmetic functions ϕ and σ
K. Ford, F. Luca, and C. Pomerance,
Bull. London Math. Soc.
, 42 (2010),
478–488.
On Carmichael numbers in arithmetic progressions
W. D. Banks and C. Pomerance,
J. Australian Math. Soc.
, 28 (2010), 313–321.
On the radical of a perfect number
F. Luca and C. Pomerance,
New York Journal of Math.
, 16 (2010), 23–30.
On the asymptotic effectiveness of Weil descent attacks
K. Karabina,
A. Menezes, C. Pomerance, and I. Shparlinski,
J. Math. Crypt.
, 4 (2010),
175–191.
Fixed points for discrete logarithms
M. Levin, C. Pomerance, and K. Soundararajan, ANTS IX Proceedings,
LNCS
6197
(2010), 6–15.
Remarks on the Pólya–Vinogradov inequality
C. Pomerance,
Integers
(Proceedings of the Integers Conference,
October 2009), 11A (2011), Article 19, 11pp.
Fibonacci integers
F. Luca, C. Pomerance, and S. Wagner,
J. Number Theory
131 (2011),
440–457.
On composite integers
for which ϕ(
)|
–1
F. Luca and C. Pomerance,
Boletin de la Sociedad
Matemática Mexicana
17 (2011), 13–21.
Primitive sets with large counting functions
G. Martin and C. Pomerance,
Publ. Math. Debrecen
, 77 (2011), 521–530.
Multiplicative properties of sets of residues
C. Pomerance and A. Schinzel,
Moscow J. Combinatorics and Number
Theory
, 1 (2011), 52–66.
On numbers
dividing the
th term of a linear recurrence
J. J. Alba González, F. Luca, C. Pomerance, and I. E. Shparlinski,
Proc. Edinburgh Math. Soc.
, 55 (2012), 271–289.
Prime-perfect numbers
P. Pollack and C. Pomerance,
Integers
(Selfridge memorial issue),
12A (2012), A14, 19 pp.
Infinitude of elliptic Carmichael numbers
A. Ekstrom, C. Pomerance, and D. S. Thakur,
J. Australian Math. Soc.
92 (2012), 45–60.
Product-free sets with high density
P. Kurlberg, J. C. Lagarias, and C. Pomerance,
Acta Arith.
155 (2012), 163–173.
The average order of elements in the multiplicative group of
a finite field
Y. Hu and C. Pomerance,
Involve
, 5-2 (2012), 229–236.
On sets of integers which are both sum-free and product-free
P. Kurlberg, J. C. Lagarias, and C. Pomerance,
Integers
(Proceedings of the
2011 Integers Conference), 12B (2012), A4, 9 pp.
On congruences of the form
σ(
) ≡
(mod
A. Anavi, P. Pollack, and C. Pomerance,
IJNT
, 9 (2012), 115–124.
On a problem of Arnold: the average multiplicative order of a given integer
P. Kurlberg and C. Pomerance,
Algebra and Number Theory
, 7 (2013),
981–999.
Sets of monotonicity for Euler's totient function
P. Pollack, C. Pomerance, and E. Treviño,
Ramanujan J.
30 (2013), 379–398.
On the distribution of some integers related to perfect
and amicable numbers
P. Pollack and C. Pomerance,
Colloq. Math.
130 (2013), 169–182.
The maximal density of product-free sets in
P. Kurlberg, J. C. Lagarias, and C. Pomerance,
Int. Math. Res. Not. IMRN
2013 (2013) #4, 827–845
(first published online February 14, 2012 doi:10.1093/imrn/rns014).
On balanced subgroups of the multiplicative group
C. Pomerance and D. Ulmer, in
Number theory and related fields,
in memory of Alf van der Poorten
J. M. Borwein, I. Shparlinski, and W. Zudlin, eds.,
Springer Proceedings in Mathematics and Statistics
43 (2013), 253–270.
Paul Erdős and the rise of statistical thinking in
elementary number theory
P. Pollack and C. Pomerance, pp. 515–523 in
Erdős Centennial
L. Lovász, I. Z. Ruzsa, and V. T. Sós, eds., János Bolyai Math. Soc.
and Springer-Verlag, Hungary, 2013.
On primes recognizable in deterministic polynomial time
S. Konyagin and C. Pomerance, pp. 159–186
in vol. 1 of
The mathematics of Paul Erdős
second edition, R. L. Graham, J. Nesetril, and S. Butler, eds.,
Springer, New York, 2013.
(This article is identical to #111 except for the update found here.)
Variant of a theorem of Erdős on the sum-of-proper-divisors function
C. Pomerance and H.-S. Yang,
Math. Comp.
83
(2014), 1903–1913.
On the local behavior of the order of appearance in the Fibonacci sequence
F. Luca and C. Pomerance,
IJNT
10
(2014), 915–933; online as DOI: 10.1142/S1793042114500079.
On the range of Carmichael's universal
exponent function
F. Luca and C. Pomerance,
Acta Arith.
162
(2014), 289–308.
Square values of Euler's function
P. Pollack and C. Pomerance,
Bull. London Math. Soc.
46
(2014), 403–414; online as
doi: 10.1112/blms/bdt097.
On integers which are the sum of
a power of 2 and a polynomial value
F. Luca, C. Gustavo Moreira, and C. Pomerance,
Bull. Brazilian Math. Soc. (NS)
45
(2014), 559–574.
The image of Carmichael's λ-function
K. Ford, F. Luca, and C. Pomerance,
Algebra & Number Theory
8-8
(2014), 2009–2026. DOI 10.2140/ant.2014.8.2009.
On the counting function of irregular primes
F. Luca, A. Pizarro-Madariaga, and C. Pomerance,
Indag. Math.
26
(2015), 147–161,
online as
Sierpiński and Carmichael numbers
W. Banks, C. Finch, F. Luca, C. Pomerance, and P. Stănică,
Trans. Amer. Math. Soc.
367
(2015), 355–376.
Divisors of the middle binomial coefficient
C. Pomerance,
Amer. Math. Monthly
122
(2015), 636–644.
(Copyright 2015, Mathematical Association of America. All rights reserved.)
Harmonious pairs
M. Kozek, F. Luca, P. Pollack, and C. Pomerance,
IJNT
11
(2015), 1633–1651, online as
The range of the sum-of-proper-divisors function
F. Luca and C. Pomerance,
Acta Arith.
168
(2015), 187–199.
On amicable numbers
C. Pomerance,
in
Analytic number theory (in honor of Helmut Maier's 60th birthday)
M. Rassias and C. Pomerance, eds., Springer, Cham, Switzerland, 2015, pp. 321–327.
On the parity of the number of small divisors of
K. Ford, F. Luca, C. Pomerance, and J. Shallit,
in
Analytic number theory (in honor of Helmut Maier's 60th birthday)
M. Rassias and C. Pomerance, eds., Springer, Cham, Switzerland, 2015, pp. 93–100.
A note on square totients
T. Freiberg and C. Pomerance,
IJNT
11
(2015), 2265–2276,
Generating random factored Gaussian integers,
easily
N. Lebowitz-Lockard and C. Pomerance,
Math. Comp.
85
(2016), 503–516.
Some problems of Erdős on the sum-of-divisors
function
P. Pollack and C. Pomerance,
Trans. Amer. Math. Soc. Ser. B
(2016),
1–26,
On integers
for which
– 1
has divisors of every degree
C. Pomerance, L. Thompson, and A. Weingartner,
Acta Arith.
175
(2016),
225–243.
Numbers divisible by a large shifted prime and large torsion subgroups of CM elliptic curves
N. McNew, P. Pollack, and C. Pomerance,
Int. Math. Res. Not.
2017;
doi: 10.1093/imrn/rnw173.
Local behavior of the composition of the aliquot and
co-totient functions
, F. Luca and
C. Pomerance, in
Analytic number theory, modular forms and q-hypergeometric series — in honor of Krishna Alladi's 60th birthday
G. Andrews and F. Garvan, eds.,
Springer Proc. Math. Stat.
221
, Springer, Cham, 2017, pp. 477–495.
Squarefree smooth numbers and Euclidean prime generators
A. R. Booker and C. Pomerance,
Proc. Amer. Math. Soc.
145
(2017), 5035–5042.
Triangles with prime hypotenuse
S. Chow and C. Pomerance,
Research in Number Theory
(2017), Art. 21, 10 pp., http://rdcu.be/wApe .
The first function and its iterates
C. Pomerance, in
Connections in Discrete Mathematics: A Celebration of the Work of Ron
Graham
, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, 2018, pp. 125–138.
Improved error bounds for the Fermat primality test
on random inputs
J. D. Lichtman and C. Pomerance,
Math. Comp.
87
(2018), 2871–2890.
Connected components of the graph generated by power maps
in prime finite fields
C. Pomerance and I. E. Shparlinski,
Integers
18A
(special issue in honor of
Jeff Shallit), Article 16, 8 pp., 2018.
Explicit estimates for the distribution of numbers free of large prime factors
J. D. Lichtman and C. Pomerance,
J. Number Theory
183
(2018), 1–23, https://doi.org/10.1016/j.jnt.2017.08.039 .
Divisor-sum fibers
P. Pollack, C. Pomerance, and L. Thompson,
Mathematika
64
(2018), 330–342,
The aliquot constant, after Bosma and Kane
C. Pomerance,
Quarterly J. Math.
69
(2018), 915–930, https://doi.org/10.1093/qmath/hay005.
Density of singular pairs of integers
R. Nedela and C. Pomerance,
Integers
18
(2018), paper A82, 7 pp.
Eigenvalues of the Laplacian on domains with fractal boundaries
P. Pollack and C. Pomerance, in
Horizons of
Fractal Geometry and Complex Dimensions,
San Luis Obispo, June, 2016
, R. G. Niemeyer, E. P. J. Pearse, J. A. Rock,
and T. Samuel, eds., Contemporary Mathematics
Vol. 731, 2019, pp. 267–277.
Primality testing with Gaussian periods
H. W. Lenstra, Jr. and C. Pomerance,
J. European Math. Soc.
21
(2019), 1229–1269.
The reciprocal sum of the amicable numbers
H. M. Nguyen and C. Pomerance,
Math. Comp.
88
(2019), 1503–1526,
The Erdős conjecture for primitive sets
J. D. Lichtman and C. Pomerance,
Proc. Amer. Math. Soc. Ser. B
(2019), 1–14, https://doi.org/10.1090/bproc/40.
Primes in prime number races
J. D. Lichtman, G. Martin, and C. Pomerance,
Proc. Amer. Math. Soc.
147
(2019), 3743–3757,
Proof of the Sheldon conjecture
C. Pomerance and C. Spicer,
Amer. Math. Monthly
126
(2019), 688–698.
Counting integers with a smooth totient
W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski,
Quarterly J. Math.
70
(2019), 1371–1386. Also
see
Progress towards a nonintegrality conjecture
S. Laishram, D. López-Aguayo, C. Pomerance, and T. Thongjunthug,
European J. Math.
(2019). https://doi.org/10.1007/s40879-019-00353-4.
Symmetric primes revisited
W. Banks, P. Pollack, and C. Pomerance,
Integers
19
(2019),
#A54, 7pp.
Counting elliptic curves with an isogeny of
degree three
M. Pizzo, C. Pomerance, and J. Voight,
Proc. Amer. Math. Soc., Ser. B
(2020),
28–42. https://doi.org/10.1090/bproc/45.
(For a missing reference, see
On the equation ϕ(
) = ϕ(
+1)
P. Kinlaw, M. Kobayashi, and C. Pomerance,
Acta Arith.
196
(2020), 69–92.
Phi, Primorials, and Poisson
P. Pollack and C. Pomerance,
Illinois J. Math.
64
(2020), 319–330.
A generalization of primitive sets and
a conjecture of Erdős
T. H. Chan, J. D. Lichtman, and C. Pomerance,
Discrete Analysis
2020:16, 13 pp.
Some thoughts on pseudoprimes
C. Pomerance and S. S. Wagstaff, Jr.,
Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques
154
No. 46 (2020), 53–72.
Long gaps in sieved sets
K. Ford, S. Konyagin, J. Maynard, C. Pomerance, and T. Tao,
J. European
Math. Soc.
23
(2021), 667–700.
DOI: 10.4171/JEMS/1020 (Published online: 2020-11-15).
Corrigendum
JEMS
25
(2023), 2483–2485.
Elliptic curves with Galois-stable cyclic
subgroups of order 4
C. Pomerance and E. Schaefer,
Research in Number Theory
, Article number: 35 (2021). Available at https://rdcu.be/cj2KY .
A note on Carmichael numbers in residue classes
C. Pomerance,
Integers
21A
(2021), The Ron Graham Memorial Volume, Article 19, 7 pp.
Algorithms for the multiplication table problem
R. Brent, C. Pomerance, D. Purdum, and J. Webster,
Integers
21
(2021), #A92, 19 pp.
On the critical exponent for
-primitive sets
T. H. Chan, J. D. Lichtman, and C. Pomerance,
Combinatorica
, DOI: 10.1007/s00493-021-4695-2.
Cyclotomic coincidences
C. Pomerance and S. Rubinstein-Salzedo,
Exp. Math.
31
(2022), 596–605.
.DOI:10.1080/10586458.2019.1660741, published online 19 September, 2019.
On primes and practical numbers
C. Pomerance and A. Weingartner,
Ramanujan J.
57
(2022),
981–1000.
DOI https://doi.org/10.1007/s11139-020-00354-y, published online Feb. 15, 2021.
Coprime matchings
C. Pomerance,
Integers
22
(2022), #A2, 9 pp.
On a nonintegrality conjecture
F. Luca and C. Pomerance,
European J. Math.
(2022), 634–639.
The man who loved problems: Richard K. Guy
, A. Granville and
C. Pomerance,
Notices of the AMS
69
(2022), 574–585.
Coprime permutations
C. Pomerance,
Integers
22
(2022), #83, 20 pp.
The denominators of the Bernoulli numbers
C. Pomerance and S. S. Wagstaff, Jr.,
Acta Arith.
209
(2023), 1–15.
Permutations with arithmetic constraints
C. Pomerance, in "Number theory in memory of Eduard Wirsing", edited by
H. Maier, J. Steuding, and R. Steuding, Springer, Switzerland, 2023, pp. 285–298.
An inequality related to the sieve of Eratosthenes
K. (S.) Fan and C. Pomerance,
J. Number Theory
254
(2024),
169–183.
Shifted-prime divisors
K. (S.) Fan and C. Pomerance, preprint, January 18, 2024.
(Current version from May 10, 2024.) To appear in a Springer volume
dedicated to Helmut Maier on his 70th birthday and edited by
J. Friedlander, C. Pomerance, and M. Rassias.
Cyclotomic primes
C. Pomerance,
J. Number Theory
276
(2025), 198–208.
(Printed version available until July 1, 2025: https://authors.elsevier.com/a/1l4o6,WUBIsSg . Also available at libraries and perhaps in a few years by Elsevier.)
Correction to: On Robin's inequality
, F. Luca,
C. Pomerance, and P. Solé,
Integers
A8
(Corrigendum, 2025), 4 pp.
Counting solvable
-unit equations and
linear recurrence sequences with zeros
A. Ostafe, C. Pomerance, and I. Shparlinski, Proc. Amer. Math. Soc.,
Article electronically published on January 16, 2026
Patterns for cyclic numbers
, C. Pomerance,
Integers (Proceedings of the 2025 Integers Conference), to appear.
A generalization of primitive roots
, S. Li and C. Pomerance, submitted for publication, April 7, 2025.
(There is some overlap between this survey article and #131.)
Fermat pseudoprimes
, S. Li and C. Pomerance,
Acta Arith.
222
(2026), 191–196.
Exceptions to the Erdős—Straus—Schinzel conjecture
, C. Pomerance and A. Weingartner, Ramanujan J. (2026) 69:31, https://doi.org/10.1007/s11139-025-01312-2.
Posted at https://rdcu.be/eY3e5
Remarks on the middle binomial coefficient
, C. Pomerance, Integers
26
(2026), paper A47, 5 pp.
Matchable numbers
, N. McNew and
C. Pomerance, preprint, April, 2026.
Last modified April, 2026.