اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDid Hypatia Know about Negative Numbers?
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In this Letter we comment on one particular aspect of Hypatias enigmatic biography by translating into English a short poem that appeared in a recent review of the third revised Polish edition of Maria Dzielskas book about Hypatia. It poses a simple and specifc question: did Hypatia know about the negative numbers?
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Translated from the Latin original Novae demonstrationes circa resolutionem numerorum in quadrata (1774). E445 in the Enestrom index. See Chapter III, section XI of Weils Number theory: an approach through history. Also, a very clear proof of the fou
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This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ sign
um negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
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