**Authors:** Germán Paz

This paper is a LaTeX document which combines previously posted papers 'Infinitely Many Prime Numbers of the Form ap±b' (viXra:1202.0063 submitted on 2012-02-18 19:13:46; url: http://vixra.org/abs/1202.0063) and 'Solution to One of Landau's Problems' (viXra:1202.0061 submitted on 2012-02-18 21:49:14; url: http://vixra.org/abs/1202.0061) into one paper. The information contained in this paper is the same as the information contained in those two original papers. No new information or results are being added. ABSTRACT. In this paper it is proved that for every positive integer 'k' there are infinitely many prime numbers of the form n^2+k, which means that there are infinitely many prime numbers of the form n^2+1. In addition to this, in this document it is proved that if 'a' and 'b' are two positive integers which are coprime and also have different parity, then there are infinitely many prime numbers of the form ap+b, where 'p' is a prime number. Moreover, it is also proved that there are infinitely many prime numbers of the form ap-b. In other words, it is proved that the progressions ap+b and ap-b generate infinitely many prime numbers. In particular, all this implies that there are infinitely many prime numbers of the form 2p+1 (since the numbers 2 and 1 are coprime and have different parity), which means that there are infinitely many Sophie Germain Prime Numbers. This paper also proposes an important new conjecture about prime numbers called 'Conjecture C'. If this conjecture is true, then Legendre's Conjecture, Brocard's Conjecture and Andrica's Conjecture are all true, and also some other important results will be true.

**Comments:** 44 Pages. Although the results that are considered as main results have not been published, some of the theorems in this paper appear in Gen. Math. Notes. See: On the Interval [n,2n]: Primes, Composites and Perfect Powers, Gen. Math. Notes, 15(1) (2013), 1-15.

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[v1] 2012-05-19 16:01:41

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