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An unconditional proof of the in…nitude of certain primes related to perfect squares Zeraoulia Elhadj Department of Mathematics, University of Tébessa, (12002), Algeria. Email:
[email protected] and
[email protected]. In this paper, a positive answer is given to the following Landau’s problem: Are there in…nitely many primes p such that p 1 is a perfect square? the proof is based an a new formula for prime numbers and the construction of two adjacent sequences. Keywords: Landau’s problem, perfect square, new prime formula, adjacent sequences. AMS 2010 Mathematics Subject Classi…cation. Primary 11A41, 11A25, Secondary 11A51, 11B34. 1. Introduction Let (pk )k 1 denotes the sequence of odd primes. The problem of representing prime numbers by some special forms of functions is an old problem. Some examples of this situation can be found in [1-2-3-4-5-6]. The well known list of Landau’s problems [7-9-10] contain the following conjecture: Conjecture 1 Are there in…nitely many primes of the form m2 + 1? A detailed historical discussion of Conjecture 1 was given by Pintz in [11]. One of the important results in this direction is that Conjecture 1 admit a generalization (together with the famous twin primes conjecture), formulated by A. Schinzel [12]: Conjecture 2 if f1 ; :::; fk are irreducible polynomials in Z [X] and their product does not have a …xed factor, then for in…nitely many integers n all values fi (n) are prime. In [13] Bateman and Horn formulated a quantitative form of Conjecture 1 based on integral formula. Note that the special case fi (x) = x + hi ; hi 2 Z of Conjecture 2 was formulated in [14] by L. E. Dickson in 1904 and the quantitative version was given in [15] by Hardy and Littlewood in 1923. By these considerations, it is clear that Conjecture 1 is the simplest case of Schinzel’s Conjecture 2 if k = 1 and deg f > 1 (the degree of f ). By using this fact some important results were reported in the current literature [16-17-18-19-20-21-22-23-24-25]. In this paper, we will propose a new formula for prime numbers. Then by using the construction of two adjacent sequences we proves that there are in…nitely many primes p such that p 1 is a perfect square.
2 2. Preliminary results Let x 2 R; then bxc denotes the ‡oor function, i.e., the largest integer not greater than x: We have bxc = j , j
(1)
x
