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- W4366180417 abstract "<abstract><p>Let $ k, l, m_1 $ and $ m_2 $ be positive integers and let both $ p $ and $ q $ be odd primes such that $ p^k = 2^{m_1}-a^{m_2} $ and $ q^l = 2^{m_1}+a^{m_2} $ where $ a $ is a positive integer with $ aequiv {pm 3}pmod 8 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je$ acute{s} $manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:</p> <p><disp-formula> <label/> <tex-math id=FE1> begin{document}$ frac{q^{2l}-p^{2k}}{2}, p^kq^l, frac{q^{2l}+p^{2k}}{2}. $end{document} </tex-math></disp-formula></p> <p>We also prove that Je$ acute{s} $manowicz' conjecture holds for non-primitive Pythagorean numbers:</p> <p><disp-formula> <label/> <tex-math id=FE2> begin{document}$ nfrac{q^{2l}-p^{2k}}{2}, np^kq^l, nfrac{q^{2l}+p^{2k}}{2}, $end{document} </tex-math></disp-formula></p> <p>for any positive integer $ n $ if for $ a = a_1a_2 $ with $ a_1equiv 1 pmod 8 $ not a square and $ gcd(a_1, a_2) = 1 $, then there exists a prime divisor $ P $ of $ a_2 $ such that $ left(frac{a_1}{P}right) = -1 $ and $ 2|m_1, aequiv 5 pmod 8 $ or $ 2not|m_2, aequiv 3pmod 8 $.</p></abstract>" @default.
- W4366180417 created "2023-04-19" @default.
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- W4366180417 date "2023-01-01" @default.
- W4366180417 modified "2023-10-01" @default.
- W4366180417 title "On the conjecture of Je$ acute{textbf{s}} $manowicz" @default.
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- W4366180417 doi "https://doi.org/10.3934/math.2023728" @default.
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