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A Formula for PrimesConsider a polynomial F(x) = x2 + x + 41. Let's check its values for a few first integers: F(1) = 43 which is prime. F(2) = 47 which is also prime. Furthermore, F(3) = 53, F(4) = 61, F(5) = 71, F(6) = 83, F(7) = 97, F(8) = 113, F(9) = 131, all of which are prime. Is it right to conclude that F(x) is a prime for all integer x? Let's check a couple more values: F(10) = 151 is a prime; F(11) = 173 and F(12) = 197 are both prime. However, it's wrong to conclude that F(x) is prime for all integer x. In fact, F(40) = 40*40 + 40 + 41 = 40*(40 + 1) + 41 = 412. Still, it's interesting that F(x) is prime for all integers from 1 through 39. G(x) = x2 - x + 41 is prime for x from 1 through 40, and H(x) = x2 - 79x + 1601 is prime for x from 1 through 80. H(81) = 41*43. 80 is a long run of primes indeed. R.K.Guy gives more examples where patterns seem to appear when looking at several small values of a variable. In some cases patterns are indeed real and valid for other values of the variable; in most cases, as above, they are figments. Guy formulates the Strong Law of Small Numbers:
These examples may serve as an introduction into the method of Mathematical Induction which consists of two steps. The first is verifying a fact for one value of a variable, say, n. The second is assuming the fact true for an arbitrary value n = k and, on this foundation, proving it for n = k+1. The second step is quite necessary. As examples above demonstrate, verifying a fact for even a large number of particular cases, does not in itself prove the fact in the general case. There is another interesting example: 4! - 3! + 2! - 1! = 19 5! - 4! + 3! - 2! + 1! = 101 6! - 5! + 4! - 3! + 2! - 1! = 619 7! - 6! + 5! - 4! + 3! - 2! + 1! = 4421 8! - 7! + 6! - 5! + 4! - 3! + 2! - 1! = 35,899 Of which all are prime. However, the very next one is composite since 326,981 = 79*4139.
References
Copyright © 1996-2009 Alexander Bogomolny |
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