Results 1 
5 of
5
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
On Wendt's Determinant and Sophie Germain's Theorem
, 1993
"... this paper belongs to a continuing line of investigations that may prove fruitful in spite of the recent announcement by Wiles of his proof of Fermat's Last Theorem. It is not unreasonable to hope for a more elementary proof than Wiles'. 1. INTRODUCTION ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
this paper belongs to a continuing line of investigations that may prove fruitful in spite of the recent announcement by Wiles of his proof of Fermat's Last Theorem. It is not unreasonable to hope for a more elementary proof than Wiles'. 1. INTRODUCTION
“Voici ce que j’ai trouvé:” Sophie Germain’s grand plan to prove Fermat’s Last Theorem
, 2010
"... A study of Sophie Germain’s extensive manuscripts on Fermat’s Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a fullyfledged, highly ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
A study of Sophie Germain’s extensive manuscripts on Fermat’s Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a fullyfledged, highly developed, sophisticated plan of attack on Fermat’s Last Theorem. The supporting algorithms she invented for this plan are based on ideas and results discovered independently only much later by others, and her methods are quite different from any of Legendre’s. In addition to her program for proving Fermat’s Last Theorem in its entirety, Germain also made major efforts at proofs for particular families of exponents. The isolation Germain worked in, due in substantial part to her difficult position as a woman, was perhaps sufficient that much of this extensive and impressive work may never have been studied and understood by anyone.
Finding integers k for which a given Diophantine Equation has no solution in kth powers of integers
"... : For a given polynomial f we use `local' methods to find exponents k for which there are no nontrivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 1. Introduction For a given polynomial f(X 1 ; X 2 ; : : : ; Xn ) 2 Z[X 1 ; X 2 ..."
Abstract
 Add to MetaCart
: For a given polynomial f we use `local' methods to find exponents k for which there are no nontrivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 1. Introduction For a given polynomial f(X 1 ; X 2 ; : : : ; Xn ) 2 Z[X 1 ; X 2 ; : : : ; Xn ] we shall investigate the set T (f) of exponents k for which the Diophantine equation (1) f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 has solutions in nonzero integers x 1 ; x 2 ; : : : ; xn . For homogenous diagonal f of degree one, Davenport and Lewis showed that k 2 T (f) whenever (n \Gamma 1) 1=2 k 18; however, Ankeny and Erdos [AE] showed that T (f) has zero density in the set of all positive integers provided that all distinct subsets of the set of coefficients of f have different sums. For general polynomials f , Ribenboim [R] showed that certain values of k cannot belong to T (f ), and the result of Ankeny and Erdos shows that T (f) has zero density, under the sam...
THE POWER OF POWERFUL NUMBERS
, 1986
"... ABSTRACT. In this note we discuss recent progress concerning powerful numbers, raise new questions and show that solutions to existing open questions concerning powerful numbers would yield advancement of solutions to deep, longstanding problems such as Fermat’s Last Theorem. This is primarily a su ..."
Abstract
 Add to MetaCart
ABSTRACT. In this note we discuss recent progress concerning powerful numbers, raise new questions and show that solutions to existing open questions concerning powerful numbers would yield advancement of solutions to deep, longstanding problems such as Fermat’s Last Theorem. This is primarily a survey article containing no new, unpublished results.