Mathematical Mayhem Mayhem Problems High School Problems
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چکیده
The readers of this article should be familiar with modular arithmetic, and may also recall Sophie Germain's identity. This article deals with an application of the identity in solving the equation 3 + 4 = 5. A word about the identity As the name says, Sophie Germain's identity was rst discovered by Sophie Germain. It reads a 4 + 4b = a + 4ab + 4b 4ab = (a + 2b) 4ab = (a + 2ab+ 2b)(a 2ab+ 2b) . What is interesting about this identity is that sums of even powers do not generally factor. Further, such sums factor only when the term we complete the square with is, in itself, a perfect square. Its main application in contest problem solving has, so far, often been trivial because of everyone knowing the identity. When starting to read the chapter about number theory in [1], I found that the identity was used in solving some simple problems related to factoring integers, and the result was an immediate consequence of it. At the time, I was quite sure that whenever I saw another problem involving the identity, I would solve it immediately. I was totally wrong! We present here an interesting application of the identity which is by no means obvious! The problem We propose the following problem, which was on the IMO Short List as late as 1991, and which also appears with this source in [2]. Problem. Solve 3 + 4 = 5 in non-negative integers. It is natural to try to prove that the only solutions are the well-known triple (2;2; 2) and the trivial (0; 1; 1). After starting to try to prove this, one may always change one's course if an obstacle is found. Counting modulo 4, we have ( 1) 1 (mod 4), so that x is even. Let x = 2n, so that 9 + 4 = 5. Now, counting modulo 5, we get ( 1) + ( 1) 0 (mod 5), since z 1, showing that n and y have opposite parities. We split into two cases: Copyright c 2000 Canadian Mathematical Society
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