By Ross Honsberger
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Mills proved the existence of k but its value is unknown. In 1963, B. M. Bredihin proved that f(x,y) = x 2 + y2 + 1 takes a prime value for infinitely many integral pairs (x, y). , odd 2 + even 2 + I = even). Our majn interest in this essay js a simple function f(x,y) which. for natural numbers x and y. iy. yields erery prime number, and takes the value of erery odd prime /lumber exactly once. 31 THE GENERATION OF PRIME NUMBERS 2. Wilson's Theorem. Our main result follows easily from one of the pillars of number theory-Wilson's theorem.
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. giving 57 consecutive primes for x = - 28, - 27, ... , 28. Euler's trinomial is not without other interest29 30 MATHEMATICAL GEMS IJ ing properties, too.
Center I and radius r) and a circumcircle O(R). Conversely, it is natural to ask when a given pair of nested circles are the incircle and circumcircle of a triangle. ) This old problem was solved by the great Euler (1707-1783). Denoting by s the distance between the centers I and 0 of the circles, he found that R2 = S2 + 2Rr if and only if I (r) and 0 (R) are the incircle and circumcircle of a triangle. However, given two suitable circles, appropriately placed, we still have a problem in determining the triangle itself.
Mathematical Gems II (Dolciani Mathematical Expositions No 2) by Ross Honsberger