Spirals of Primes
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|Bryan Clair, University of Chicago|
|This page will generate a spiral of numbers, with the primes marked as stars, and the composites as dots. The center of the spiral is marked with an 'X'. Specify the number at the center of the spiral below, and the width (and therefore also the height) in columns of the spiral. You should notice that the prime numbers exhibit a tendency to fall on diagonal lines. This works for any start number, but seems to work better for primes, and better still for the first of two twin primes. It is a complete mystery why this phenomenon occurs. Euler's classic polynomial x *x + x + 41 generates primes for the first 39 values of x. This polynomial corresponds exactly to the diagonal (from top left to bottom right) of the spiral when you start with 41. On this diagonal, 47% of the numbers less than ten million are prime. In general, starting with n yields the polynomial x *x + x + n along the diagonal.|
|Levels:||High School (9-12), College|
|Resource Types:||Calculators, Problems/Puzzles|
|Math Topics:||Polynomials, Prime Numbers, Patterns/Relationships|
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