The Journal of the Indian Mathematical Society

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The Equivalence of Two Conjectures in the Theory of Numrers


Affiliations
1 University of Illinois, United States
2 University of Colorado, United States
     

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The equivalence of these two conjectures can even be put in the following sharper form : If N is any positive integer, Conjecture I is true for all n ≤ N if and only if Conjecture II is true for all n ≤ N and all real characters x. In one direction this is trivial, since λ(v) = (v|N*)forv= 1,2,…, N, where (x | y) denotes the Legendre- Jacobi symbol and N* denotes the smallest positive integer such that (p | N*) = - 1 for all primes p not exceeding N.
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  • The Equivalence of Two Conjectures in the Theory of Numrers

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Authors

P. T. Bateman
University of Illinois, United States
S. Chowla
University of Colorado, United States

Abstract


The equivalence of these two conjectures can even be put in the following sharper form : If N is any positive integer, Conjecture I is true for all n ≤ N if and only if Conjecture II is true for all n ≤ N and all real characters x. In one direction this is trivial, since λ(v) = (v|N*)forv= 1,2,…, N, where (x | y) denotes the Legendre- Jacobi symbol and N* denotes the smallest positive integer such that (p | N*) = - 1 for all primes p not exceeding N.