Browsing by Author "Yang, YiFan"

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A Cubic Analogue of the Jacobsthal Identity
(2011-04-01)
Chan, Heng Huat
;
Long, Ling
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Yang, YiFan
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應用數學系
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Department of Applied Mathematics
It is well known that if p is a prime such that p I (mod 4). then p can be expressed as a sum of two squares. Several proofs of this fact are known and one of them, due to E. Jacobsthal, involves the identity p = x(2) + y(2) with x and y expressed explicitly in terms of sums involving the Legendre symbol. These sums are now known as the Jacobsthal sums. In this short note, we prove that if p equivalent to 1 (mod 6). then 3p = u(2) + uv + nu(2) for some integers u and v using an analogue of Jacobsthal's identity.