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1983 17 200 207 Kaarli, K. , On certain classes of algebras generating arithmetical affine complete

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), 935 – 974 . [3] K aptan , Deniz A. Large gaps between primes in arithmetic progressions , Not intended far publication, available at arXiv arXiv

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Wiley & Sons, Ltd., Chichester, 1988. MR 89j :60006 Algebraic probability theory Zempléni, A., On arithmetical properties of the

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. [9] Vaughan , R. C. 2005 A variance for k -free numbers in arithmetic progressions Proc. London Math. Soc., (3) 91 573 – 597 10.1112/S

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] Deza , E. , Varukhina , L. 2008 On mean values of some arithmetic functions in number fields Discrete Math. 308 4892 – 4899 10.1016/j.disc.2007.09.008 . [4] Heath-Brown , D. R

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Kaarli, K., On certain classes of algebras generating arithmetical affine complete varieties, General Algebra and Applications , 152-161. Heldermann Verlag, Research and Exposition in Mathematics, Vol. 20 (Berlin, 1993). MR 93m :08006

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Abstract  

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x 2dy 2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.

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For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|n d 2. On the other hand, we prove that for the function f(n) := ∑p|n p 2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

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Abstract  

We give a new characterization of the logarithm as an additive arithmetical function.

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BALOG, A. and PERELLI, A., Exponential sums over primes in an arithmetic progression, Proc. Amer. Math. Soc. 93 (1985), 578-582. MR 86b :11053 Exponential sums over primes in an arithmetic progression

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