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Register a new userThe Pythagorean Tree: A New Species
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H. Lee Price
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In 1967 the Dutch mathemetician F.J.M. Barning described an infinite, planar, ternary tree*. Seven years later, A. Hall independently discovered the same tree. Both used the method of uni-modular matrices to transform one triple to another. A number of rediscoveries have occurred more recently. In this article we announce the discovery of an entirely different ternary tree, and show how it relates to the one found by Barning and Hall.
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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 four squares theorem based on Eulers is Theorem 370 in Hardy and Wright, An introduction to the theory of numbers, fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isnt clear please email me.
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 $4n+1$ signum 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).
We show that the use of the main characteristics of the circle map leads naturally to establish a few statements on primes and pseudoprimes. In this way a Fermats theorem on primes and some interesting properties of pseudoprimes are obtained.
This paper is an exposition and review of the research related to the Riemann Hypothesis starting from the work of Riemann and ending with a description of the work of G. Spencer-Brown.
A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali obtained the first effective bound on the sizes of S.P numbers. Modifying Shah Alis method, we obtain an improved bound on the number of digits in a base-r S.P number. Our bound is the first sharp bound found for the case r=2.
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