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By Chowdhury K.C.

ISBN-10: 8186299653

ISBN-13: 9788186299654

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If we take this for granted, we can prove that there are infinitely many primes 4n+1. In fact we can prove THEOREM 14. There are infinitely many primes of the form 8n+5. 72... p2+22, a sum of two squares which have no common factor. The square of an odd number 2m+1 is 4m(m+1)+1 and is 8n+1, so that q is 8n+5. Observing that, by Theorem 13, any prime factor of q is 4n±1, and so 8n+1 or 8n+5, and that the product of two numbers 8n+1 is of the same form, we can complete the proof as before. All these theorems are particular cases of a famous theorem of Dirichlet.

Hence there cannot be any abnormal numbers and this is the fundamental theorem. 2. Mr. Ingham tells us that the argument used here is due to Bohr and Littlewood: see Ingham, 2. 3. For Theorems 11, 12, and 14, see Lucas, Theorie des nombres, i (1891), 353-4; and for Theorem 15 see Landau, Handbuch, 422-46, and Vorlesungen, i. 79-96. An interesting extension of Theorem 15 has been obtained by Shiu (J. London Math. Soc. (2) 61 (2000), 359-73). This says that for a and b as in Theorem 15, the sequence of primes contains arbitrarily long strings of consecutive elements, all of which are of the form an + b.

We be 1 and the regions Rp, with their boundaries, may give an exact proof as follows. Suppose that A is the area of R0, and A the maximum distance of a point of Cot from 0; and that we consider the (2n+ 1)2 regions Rp corresponding to points of A whose coordinates are not greater numerically than n. All these regions lie in the square whose sides are parallel to the axes and at a distance n + A from O. Hence (since the regions do not overlap) (2n + 1)2i (2n + 2A)2, 0< 1+ A-n+ 2 , and the result follows when we make n tend to infinity.

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A first course in theory of numbers by Chowdhury K.C.

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