科研之友
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摘要
Let a, b, c be a primitive Pythagorean triple and set a = m(2) - n(2), b = 2mn, c = m(2) + n(2), where m and n are positive integers with m > n, gcd(m, n) = 1 and m not equivalent to n (mod 2). In 1956, Jesmanowicz conjectured that the only positive integer solution to the Diophantine equation (m(2) - n(2))(x) + (2mn)(y) = (m(2) + n(2))(z) is (x, y, z) = (2, 2, 2). We use biquadratic character theory to investigate the case with (m, n) equivalent to (2, 3) (mod 4). We show that Jesmanowicz conjecture is true in this case if m + n not equivalent to 1 (mod 16) or y > 1. Finally, using these results together with Laurent's refinement of Baker's theorem, we show that Jesmanowicz' conjecture is true if (m, n) equivalent to (2, 3) (mod 4) and n < 100.
- 出版日期2017-2
- 单位海南大学
