On the integer points on some special hyper-elliptic curves over a finite field

AbstractIf lr(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ⋯ (x + r − 1)(modp) has a solution, we conjecture that lr(p) ≤ r2 − r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y2 ≡ (x + a1) ⋯ (x + ar) is suggested, where the a's are fixed positive integers

The greatest prime factor of x<SUP>2</SUP>−1

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The representation of a large number as a sum of 'almost equal'cubes

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A theorem on irrational indefinite quadratic forms

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On stronger conjectures that imply the Erd\"os-Moser conjecture

The Erd\"os-Moser conjecture states that the Diophantine equation $S_k(m) = m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers $k$ and $m$ with $k \geq 2$. We show that stronger conjectures about consecutive values of the function $S_k$, that seem to be more naturally, imply the Erd\"os-Moser conjecture.Comment: 7 page

Leudesdorf's generalization of Wolstenholme's theorem

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Improvement of a theorem of Linnik and Walfisz

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Congruence properties of partitions

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On a theorem of Walfisz

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A theorem in arithmetic

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