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Small solutions of the congruence

≡ 0 (mod p)

Published online by Cambridge University Press:  24 October 2008

Kenneth S. Williams
Kenneth S. Williams
Affiliation:
Carleton University, Ottawa, Canada
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Extract

Throughout this paper a0, a1, a2, l1, l2 denote fixed integers with l1 ≥ 2, l2 ≥ 2. We let l = max(l1, l2) and let P be the set of primes Mordell (4) has shown that for any sufficiently large prime p the congruence

is soluble. Thus there are at most a finite number of such p for which (1·1) is insoluble. If there is at least one prime p ∈ P for which (1·1) is insoluble, we let p0 denote the largest of such p, so that (1·1) is soluble for all p ∈ P with p > p0 but not for p = p0. Otherwise (1·1) is soluble for p ∈ P and we let p0 = 1. From the work of Mordell (4) we have

For p ∈ P with p > p0 (1·1) is thus always soluble and any such solution (x1, x2) can be taken to satisfy

Chalk(2) has posed the problem of estimating a ‘small’ solution of (1·1), at least for p sufficiently large; that is a solution for which p in the inequality (1·3) can be replaced by something less than p. Smith(5) has shown that for p sufficiently large there is always a solution satisfying 1 ≤ xi ≪ p¾log p (i = 1, 2). It is the purpose of this paper to prove the following sharper and more precise result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 3 , November 1971 , pp. 409 - 412
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

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