There should be more math! This could be mathier!
There should be more math! This could be mathier!
-- Buffy Summers, channeling Rupert Giles, BtVS Episode 2.8 ("The Dark Age")
An explicit approach to Hypothesis H for polynomials over
finite fields [PDF]
The
anatomy of integers.
Proceedings of a conference on the anatomy of integers, Montreal,
March 13th-17th, 2006 eds: J.M. de Koninck, A. Granville and F.
Luca, pp. 259--273
On a conjecture of Beard, O'Connell and West concerning
perfect polynomials (w/ L.
Gallardo and O. Rahavandrainy) [PDF]
Finite
Fields and their Applications 14 (2008),
no. 1, 242-249
Simultaneous prime specializations of polynomials over
finite fields [PDF]
Proceedings
of the London Mathematical Society 97
(2008), no. 3, 545-567
A polynomial analogue of the twin prime conjecture
[PDF]
Proceedings of the
AMS 136 (2008),
no. 11, 3775-3784
Arithmetic properties of polynomial specializations over
finite fields [PDF]
Acta
Arithmetica 136 (2009),
no. 1, 57-79
On the distribution of sociable numbers (w/
M. Kobayashi and C. Pomerance) [PDF]
J.
Number Theory 129,
no. 8, 1990-2009
The greatest common divisor of
a number and its sum of divisors [PDF]
Michigan
Math. J. (to
appear)
On some friends of the sociable numbers [PDF], submitted.
On polynomial analogues of the Goldbach and twin prime conjectures (w/ Andreas Bender) [PDF], submitted.
Revisiting Gauss's analogue of the prime number theorem for polynomials over a finite field [PDF], submitted.
Long gaps between deficient numbers [PDF], submitted.
Perfect numbers with identical digits [PDF], submitted.
A note on Dickson's finiteness theorem for odd perfect numbers, in preparation.
A remark on sociable numbers of odd order, in preparation.
Not
always buried deep: A second course in elementary number
theory
Published by the
American Mathematical Society. Please consult the errata list here.
These are papers that could possibly appear some day, but which probably would require substantial development first.
A note on Hilbert's solution
to Waring's problem
We show
how to derive the explicit bound g(k) < (2k+1)^{2000 k^5} from a
modification of Hilbert's original solution to Waring's problem.
This builds upon and improves work of G. J. Rieger.
An irrational proof that there
are infinitely many rational primes
We exhibit a connection
between the infinitude of primes and the irrationality of sqrt(2).
(and not intended for publication)
A proof of the Sylvester--Schur theorem [PDF]
(based
on the treatment in Narkiewicz's Classical
Problems in Elementary Number Theory)
Remarks on Hypothesis H and an impossibility theorem of Ram Murty [PDF]
An analogue of Goldbach's conjecture for certain polynomial rings [PDF]
Primes, polynomials and patterns (Talk)
[PDF]
Based
on talks at Dartmouth's 2006 Exploring Math Program and the Ross
Summer Mathematics Program 2007 reunion conference.
Rational cubic reciprocity (Talk) [PDF]
The distribution of sociable numbers (Talk) [PDF]