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Arithmetic of Elliptic Curves and Diophantine Equations
"... Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three nonzero distinct nth powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One cannot fa ..."
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Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three nonzero distinct nth powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One cannot fail to notice that it is a variant of the FermatWiles theorem. We would like to present the ideas which led H. Darmon and the author to the solution of Denes' problem in [13]. Many of them are those (due to Y. Hellegouarch, G. Frey, J.P. Serre, B. Mazur, K. Ribet, A. Wiles, R. Taylor, ...) which led to the celebrated proof of Fermat's last theorem. Others originate in earlier work of Darmon (and Ribet). The proof of Fermat's last theorem
Transformations of U(n + 1) multiple basic hypergeometric series
 UMEMURA (EDS.), PHYSICS AND COMBINATORICS: PROCEEDINGS OF THE NAGOYA 1999 INTERNATIONAL WORKSHOP (NAGOYA
, 1999
"... The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system An. Our approach to this theory employs partial fraction decompositions, qdifference equations, and suitable ..."
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The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system An. Our approach to this theory employs partial fraction decompositions, qdifference equations, and suitable multidimensional matrix inversions. These series were strongly motivated by Biedenharn and Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups U(n + 1), or equivalently An. The foundation of our theory is the U(n + 1) multiple sum renement of the terminating classical qbinomial theorem. This result contains as special, limiting, or transformed cases both the Macdonald identities for An, and U(n + 1) multiple sum extensions of the classical qbinomial theorem, Ramanujan's 1 1 sum, the balanced 3 2 summation theorem, and Rogers ' classical terminating verywellpoised 6 5 summation
Binomial Coefficients, the Bracket Function, and Compositions with Relatively Prime Summands.” The Fibonacci Quarterly Vol
, 1964
"... Bratter on the occasion of his 70th birthday. It is known [5] that a n e c e s s a r y and sufficient condition for p to be prime is that for every natural number n (1) Q 5 [  ] (modp). where Txl denotes the g r e a t e s t integer less than or equal to x. Indeed this result is equivalent to the c ..."
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Bratter on the occasion of his 70th birthday. It is known [5] that a n e c e s s a r y and sufficient condition for p to be prime is that for every natural number n (1) Q 5 [  ] (modp). where Txl denotes the g r e a t e s t integer less than or equal to x. Indeed this result is equivalent to the congruence k k (1 x) = 1 x (mod p) as is evident from the generating functions (2)
LowDimensional Lattices IV: The Mass Formula
 Proc. Royal Soc. London, A
, 1988
"... The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, ..."
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The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, unlike earlier ones, is visibly invariant under scale changes and dualizing. We use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series. We also give tables of the "standard mass", the Lseries S (n / m)m  s (m odd), and genera of lattices of determinant d 25. 1.
On the representation of unity by binary cubic forms
 Trans. Amer. Math. Soc
"... Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z ..."
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Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x 2 − xy − y 2)orx(x 2 − 2y 2). The proofs of these results rely upon the method of ThueSiegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) =1forF cubic and irreducible of positive discriminant DF ≤ 10 6. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) =m and to Mordell’s equation y 2 = x 3 + k, wherem and k are nonzero integers. 1.
The Klein quartic in number theory
, 1999
"... Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � ..."
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Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.
PERFECT POWERS: PILLAI’S WORKS AND THEIR DEVELOPMENTS
, 2009
"... Abstract. A perfect power is a positive integer of the form ax where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y ..."
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Abstract. A perfect power is a positive integer of the form ax where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation ax − by = k is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai’s work on Diophantine questions, we quote some later developments and we discuss related open problems.
Some Formulae For Partitions Into Squares
 Discrete Math
, 2000
"... . We consider the new problem of determining the number of partitions of a number into a fixed number k of squares, and find explicit formulae in the cases k = 2, 3, 4. Classification: 11P81, 05A17 Keywords: partitions, squares. 1. Introduction and statement of results We consider the problem of ..."
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. We consider the new problem of determining the number of partitions of a number into a fixed number k of squares, and find explicit formulae in the cases k = 2, 3, 4. Classification: 11P81, 05A17 Keywords: partitions, squares. 1. Introduction and statement of results We consider the problem of determining the number of partitions of a number into a fixed number of squares. This problem is distinct from, but not unrelated to, the corresponding classical problem of determining the number of representations of a number as the sum of a fixed number of squares. Whereas the classical problem has received an enormous amount of attention over the years, see Dickson [3, Vol. II, Chapters VIIX], it appears that this (restricted) partition problem has not previously been investigated. Hardy and Ramanujan [4, p.305] and Baxter [1] have looked at the problem of partitions of a number into (an unrestricted number of) squares, which finds application in statistical mechanics. To see why one m...
Regularity Properties of the Stern Enumeration of the Rationals
"... s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1 ..."
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s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3. 1