A rather old conjecture asserts that if m = p is prime then, for any fixed ε > 0 and sufficiently large p, for every integer a there are integers x and y with |x|, |y| ≤ p~(1/2+ε) and such that a ≡ xy (mod p); see [14; 16; 17; 18] and references therein. The question has probably been motivated by the following observation. Using the Dirichlet pigeon-hole principle, one can easily show that, for every integer a, there exist integers x and y with |x|, |y| ≤ 2p~(1/2) and with a ≡ y/x (mod p). Unfortunately, this is known only with |x|, |y| ≥ Cp~(3/4) for some absolute constant C > 0, which is due to Garaev.
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机译：一个相当古老的猜想认为，如果m = p为素数，则对于任何固定的ε> 0和足够大的p，对于每个整数a，都有x和y带有| x |，| y |的整数。 ≤p〜（1/2 +ε）且that xy（mod p）;见[14; 16; 17; 18]和其中的参考文献。该问题可能是由于以下观察所致。使用狄里克雷信鸽原理，可以很容易地证明，对于每个整数a，都有x和y以及| x |，| y |的整数。 ≤2p〜（1/2）且y / x为（mod p）。不幸的是，这仅在| x |，| y |中知道。对于某些绝对常数C> 0≥Cp〜（3/4），这是由于Garaev引起的。
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