# Gica - ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36...

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Unformatted text preview: ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36, Number 6, 2006 QUADRATIC RESIDUES OF CERTAIN TYPES ALEXANDRU GICA ABSTRACT. The main purpose of the paper is to show that if p is a prime different from 2 , 3 , 5 , 7 , 13 , 37, then there exists a prime number q smaller than p , q ≡ 1 (mod 4), which is a quadratic residue modulo p . Also, it is shown that if p is a prime number which is not 2 , 3 , 5 , 7 , 17, then there exists a prime number q ≡ 3 (mod 4), q < p , which is a quadratic residue modulo p . 1. Introduction. In [ 2 ] it is shown that any n ∈ N , n > 3, could be written as n = a + b, a, b being positive integers such that Ω( ab ) is an even number. If m ∈ N , m ≥ 2, has the standard decomposition m = p a 1 1 · p a 2 2 ··· p a r r then the length of m is Ω( m ) = ∑ n i =1 a i . We put Ω(1) = 0. In connection with the above quoted result, the following open problem naturally arises. Open problem. What numbers n can be written as n = a 2 + b , where a, b are positive integers, the length of b being an even number? Trying to solve this problem was the starting point for the main result of this paper. Theorem 1. Let p be a prime number p 6 = 2 , 3 , 5 , 7 , 13 , 37 . There exists a prime number q such that q < p , q ≡ 1 (mod 4) and ( q/p ) = 1 . We will prove also a similar result which has, however, an elementary proof: 2000 AMS Mathematics Subject Classification. Primary 11A15, 11E25, 11R29. Key words and phrases. Quadratic residue, length, numerus idoneus. Received by the editors on March 22, 2004, and in revised form on April 9, 2004. Copyright c 2006 Rocky Mountain Mathematics Consortium 1867 1868 A. GICA Theorem 2. If p is a prime not equal to 2 , 3 , 5 , 7 , 17 , then there exists a quadratic residue modulo p , where q < p and q ≡ 3 (mod 4) . We have to mention that finding the properties of n ( p ), the least prime number which is quadratic residue modulo a prime p , is a classical problem. We quote here [ 6 ] where it is shown that n ( p ) = O ( p α ) , where α is a fixed real number for which α > 1 / 4 e − 1 / 2 ....
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Gica - ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36...

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