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Number Theory - Fermat's Failed Prime Number Formula

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Many mathematicians, including Mersenne and Euler, have tried to find a formula that will define all the prime numbers. No one has ever succeeded.

Fermat had one of the most famous failures. He thought that if he squared 2 and then raised the square of 2 to a higher power, which he labeled n (a whole number), then the results would be nothing but primes. His formula looks like this: 2n 2 + 1 = a prime number. This formula appeared to work until Leonhard Euler proved it wrong. Euler found that if 5 is substituted for n in the formula 22n + 1, the resulting number is 4,294,967,297, which can be divided equally by 641 and 6,700,417.


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