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By Adrian I. Ban, Sorin G. Gal

Introduces a mode of study which might be utilized in quite a few fields of arithmetic. Examines, in a scientific means, the quantitative characterizations of the deviation from a estate, referred to as the illness of a estate.

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This idea was first used by Heath-Brawn [15], and for problems of Waring's type, by the first author [3]. k=l has solutions for all large natural numbers n. This has attracted many writers since it was first treated by Roth [25] with s = 50. The current record s = 14 is due to Ford [8]. 8) in primes. This is explicitly mentioned in Thanigasalam [27] where it is shown that when s = 23 there are prime solutions for all large odd n. An improvement of this result may well be within reach, and we intend to return to this topic elsewhere.

7), whence IS;(p, a ) [ fi+ we have < - for kl = 4 and 5. When p = 7 and kl = 3, we can check by hand again that M(7, n ) > 0 unless n 5 (mod 7). Collecting all the conclusions, we obtain the lemma. We next turn to the singular series which occur in our quaternary problems. In such circumstances, we set s = 2. 7. 3) with s = 2 and natural numbers k and kj (0 j 2) which are less than 6 , and suppose that min{k, ko) = 1. Then the infinite series Gd(n) = Ad(q, n ) converges absolutely, and one has < < as well as Proof.

Jia and K. Analytic Number Theory, 93-98. 94 A generalization of E. Lehmer's congruence and its applications ANALYTICNUMBER THEORY where 95 Corollary 2. If n is odd, then (n- is Fermat's quotient, using (1) and other similar congruences, he obtained various criteria for the first case of Fermat's Last Theorem (Cf. [8]). The proof of (1) followed the method of Glaisher 121, which depends on Bernoulli polynomials of fractional arguments. In this paper, we follow the same way t o generalize (1) to modulo arbitary positive integers, however, we need establish special congruences concerning the quotients of Euler.

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