M. Cowles

On the number of zeros of diagonal cubic forms

Abstract Let M s , be the number of solutions of the equation X 1 3 + X 2 3 + … + X s 3 =0 in the finite field GF ( p ). For a prime p ≡ 1(mod 3), ∑ s=1 ∞ M s X s = x 1 − px + x 2 (p − 1)(2 + dx) 1 − 3px 2 − pdx 3 , M 3 = p 2 + d(p − 1) , and M 4 = p 2 + 6(p 2 − p) . Here d is uniquely determined by 4 p = d 2 + 27b 2 and d ≡ 1( mod 3) .

On the divisibility of the class number of imaginary quadratic fields

Abstract The theorem presented in this paper provides a sufficient condition for the divisibility of the class number of an imaginary quadratic field by an odd prime. Two corollaries to this theorem are also included. They represent special cases of the theorem which in general use are somewhat easier to apply.

The least prime quadratic residue and the class number

For an odd prime p, let l(p) denote the least positive prime which is a quadratic residue modulo p. An elementary proof is given of the following: For p ≡ 3 mod 8 and p > 3, l(p) = (p + 1)4 iff the class number h(−p) = 1.

On the number of conjugacy classes in SL(2,Z)

Let H(t) be the number of conjugacy classes of elements in SL(2, L) with trace t, and let h(n) be the number of equivalence classes of binary quadratic forms with discriminant n. Then for t≠±2, H(t)=h(t2−4). For all real θ > 0 there is a T(θ) such that whenever |t|>T(θ), H(t)>|t|1−θ. There is a c>0 such that for those t such that t2−4 is squarefree, H(t)≤c|t|.

On Y2 = X3 + k and the Thue rank of cubic curves

Abstract The intersection of cords and tangents with the curve Y 2 = X 3 + k is exploited to locate additional lattice points and to define the Thue rank of the curve.

Remarks on Equations Related to Fermat’s Last Theorem

For odd k, define θ(k) as the least value of s such that has a non-trivial Solution over the integers. Fermat’s Last Theorem impl ies that θ(k) > 3 for odd k > 3.

On x3 + y3 = D

Abstract The simplest case of Fermats last theorem, the impossibility of solving x3 + y3 = z3 in nonzero integers, has been proved. In other words, 1 is not expressible as a sum of two cubes of rational numbers. However, the slightly extended problem, in which integers D are expressible as a sum of two cubes of rational numbers, is unsolved. There is the conjecture (based on work of Birch, Swinnerton-Dyer, and Stephens) that x3 + y3 = D is solvable in the rational numbers for all square-free positive integers D ≡ 4 (mod 9). The condition that D should be square-free is necessary. As an example, it is shown near the end of this paper that x3 + y3 = 4 has no solutions in the rational numbers. The remainder of this paper is concerned with the proof published by the first author (Proc. Nat. Acad. Sci. USA., 1963) entitled “Remarks on a conjecture of C. L. Siegel.” This pointed out an error in a statement of Siegel that the diophantine equation ax3 + bx2y + cxy2 + dy3 = n has a bounded number of integer solutions for fixed a, b, c, d, and, further, that the bound is independent of a, b, c, d, and n. However, x3 + y3 = n already has an unbounded number of solutions. The paper of S. Chowla itself contains an error or at least an omission. This can be rectified by quoting a theorem of E. Lutz.