Number Theory

Topics in Number Theory by J. S. Chahal (auth.)

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By J. S. Chahal (auth.)

This e-book reproduces, with minor alterations, the notes ready for a direction given at Brigham younger collage through the educational yr 1984-1985. it really is meant to be an advent to the idea of numbers. The viewers consisted mostly of undergraduate scholars with out extra historical past than highschool arithmetic. The presentation used to be hence stored as uncomplicated and self-contained as attainable. although, as the dialogue was once, in most cases, carried a long way adequate to introduce the viewers to a couple components of present study, the publication also needs to be precious to graduate scholars. the one prerequisite to interpreting the booklet is an curiosity in and flair for mathe­ matics. notwithstanding the subjects could appear unrelated, the research of diophantine equations has been our major objective. i'm indebted to a number of mathematicians whose released in addition to unpublished paintings has been freely used all through this publication. particularly, the Phillips Lectures at Haverford university given through Professor John T. Tate were a massive resource of fabric for the publication. a few elements of bankruptcy five on algebraic curves are, for instance, in accordance with those lectures.

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Thus g and (p2 -1)/8 have the same parity and o For the proof of the law of quadratic reciprocity, we need the following combinatorial result. 9. -. q 2 2 40 Chapter 3 If we put PROOF. S(p, q) = L (p-J)/2 j=i [jq] -, P we must show that s(p,q)+s(q,p)= (p-1)(q-1) 4 . 11) It is easy to see that for each j = 1, ... , (p - 1)/2, [jq/ p] is the number of integers in the open interval (O,jq/p) = {x E IRlo < x

2. Find all the primes p for which p) = 1. 3. Find all the primes p for which (S/p) =-1. 3. 18) represents every non-negative integer. 19) which shows that if nj is a sum of two squares, j = 1,2, then so is ntn2' This reduces the task of proving the above statement to the proof of the fact that every prime p is a sum of four squares. 11 (Lagrange). Every positive integer is a sum of four squares. PROOF (Euler). 20) it suffices to show that every odd prime p is a sum of four squares. We shall prove it in two steps: 1.

Xk, P - YI, ... ,p - Yg are all distinct elements of f p • To prove this all we have to show is that Xi ¥- P - Yj for all i, j. , Xi = P - Yj for some i, j. Noting that p is zero in f p (and performing the operations in f p) we get Xi + Yj = O. But Xi = ar and Yj = as for some r, s (1 :5 r, s:5 (p - 1)/2). Hence a(r + s) = o. , pi r + s. But this is impossible because 2:5 r + s :5 P - 1. 9) now imply that the two sets X and Z are equal and hence (in f p) p-l 1·2· .. Xk(P - YI) ... XkYI·· ·Yg = P -1 (-1) ga .

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