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Constructing x 2 for primes p = ax 2 + by 2

By Zhi-hong Sun


Let a and b be positive integers and let p be an odd prime such that p = ax2 +by2 for some integers x and y. Let λ(a, b; n) be given by q ∏∞ k=1 (1−qak) 3 (1−q bk) 3 = ∑∞ n=1 λ(a, b; n)qn. In this paper, using Jacobi’s identity ∏∞ n=1 (1−qn) 3 = ∑∞ k=0 (−1)k(2k+1)q k(k+1) 2, we construct x2 in terms of λ(a, b; n). For example, if 2 ∤ ab and p ∤ ab(ab+1), then (−1) a+b 2 x+ b+1 2 (4ax 2 −2p)

Topics: Binary quadratic form, Jacobi’s identity
Year: 2013
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