Browsing by Author "Long, Ling"

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A Cubic Analogue of the Jacobsthal Identity
(2011-04-01)
Chan, Heng Huat
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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.
A SHORT PROOF OF MILNE'S FORMULAS FOR SUMS OF INTEGER SQUARES
(2005-12-01)
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
We give a short proof of Milne's formulas for sums of 4n(2) and 4n(2) + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.
Jacobsthal identity for Q(root-2)
(2012-11-01)
Hashimoto, Ki-ichiro
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Long, Ling
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Yang, Yifan
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應用數學系
;
Department of Applied Mathematics
Let p be a prime congruent to 1 or 3 modulo 8 so that the equation p = a(2) + 2b(2) is solvable in integers. In this paper, we obtain closed-form expressions for a and b in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.