We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Speci cally, we show that the structure theorem for nite abelian groups is provable in ), and use it to derive Fermat's little theorem and Euler's criterion for the Legendre
Bounds of quadratic residues in arithmetic progression
β Scribed by Sahib Singh
- Publisher
- Elsevier Science
- Year
- 1970
- Tongue
- English
- Weight
- 255 KB
- Volume
- 2
- Category
- Article
- ISSN
- 0022-314X
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π SIMILAR VOLUMES
Let G(k, r) denote the smallest positive integer g such that if 1=a 1 , a 2 , ..., a g is a strictly increasing sequence of integers with bounded gaps a j+1 &a j r, 1 j g&1, then [a 1 , a 2 , ..., a g ] contains a k-term arithmetic progression. It is shown that G(k, 2) > -(k & 1)Γ2 ( 43 ) (k&1)Γ2 ,
proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, ..., N} is at least cN 1/4 , and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N] d