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By Ross Honsberger

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42 + 14, 42 == 3 . 14. , divisions) in an appJication of the Euclidean algorithm never exceeds 5 times the number of digits in the lesser number. Having wondered about this now and again for a long time, I was delighted recently to run across a beautiful proof by H. Grossman in the American Mathematical Monthly, Volume 3], )924, page 443. Later I discovered the same proof in Sierpinski's outstanding Theory of Numbers (2]. Suppose an is the lesser number in a pair (an+ I' an) for which the Euclidean algorithm contains n steps (n > I).

The proof is not difficult and one can scarcely remain untouched by the delightful power which mathematics displays in this neat application. This is the discovery of John Conway, of Cambridge. We begin by noting that if one were able to reach a particular lattice point P on level five, one could, by doing the same thing from a starting position farther to the left or to the right, arrive at any specified lattice point on level five. It's all or nothing for level five. We settle the matter by showing that an arbitrary but definite lattice point P on level five is inaccessible.

Thus x 2 + x + 41 yields primes for the eighty consecutive integers x = - 40. - 39, ... , 38, 39. Equivalently, the function f(x - 40) = x 2 - 79x + 1601 gives these eighty prime values for x = 0, I. • 79. At present this shares the record for the longest string of consecutive integers for which a quadratic yields prime numbers exclusively. (The function x 2 - 2999 x + 2248541 also yields 80 primes for x = 1460, 1461, ... ) Little competition is provided by 6x 2 + 6x + 3]. • 28. The binomial 2x2 + 29 does better.

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