Ulam spiral

In mathematics, the Ulam spiral, or prime spiral is a simple method, primes represent graphically. It was discovered in 1963 by the Polish mathematician Stanislaw Ulam Marcin during a scientific lecture, as he scribbled out of boredom rows of numbers on a paper. He began with a "1 " in the middle, and then continued in a spiral form:

Then he circled all the prime numbers and received the following pattern:

To his surprise, almost all primes were on diagonal lines, as the graph on the right shows. This is a Ulam Spiral of size 200 × 200, where the primes are marked by black dots. The diagonal lines are clearly visible.

It seems as if the diagonal lines always appear, regardless of the size of the spiral. This seems to be the case even when the initial number is much greater than 1. It follows that there are many integers a, b and c, with which the function

Results in a very large amount of prime numbers. In March 1964, the Ulam Spiral is depicted on the cover of the magazine Scientific American.

At sufficiently large distance from the center, you can also discover horizontal and vertical lines.

The prime researchers were these figures have long been aware. In the 18th century the Swiss mathematician Leonhard Euler had discovered the formula, each of which resulted primes for consecutive values ​​between 0 and 15. In fact, these 16 figures those which also appear in Ulam diagram on the main diagonal are: 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227 and 257 later Euler found another formula that yielded exclusively for prime numbers between 0 and 40. By recalculating the computer showed that this second Euler's formula was surprisingly good, as it gives prime numbers for up to 10 million in 22.08 % of cases. Ulam was more formulas whose percentages in the generation of prime numbers were almost as good as the Euler formula. The pattern of the Ulam spiral can not be fully explained, however, until today.