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Prime Numbers: Leonhard Euler
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During the beginning of the 17th century, Fermat proved that every prime number of the form 4n + 1 could be written as the sum of two squares. Fermat ... stated that the numbers 2n + 1 are always prime if n is a power of 2. Numbers that have this property were called Fermat numbers. After 100 years, Euler prooved that this formula does not always work because 232 + 1 is equal to 4,294,967,297, which is not prime (it is divisible by 641).
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Euler's most famous prime generator is x2 + x + 41. Starting a spiral at 41 produces a grid with an amazing, unbroken sequence of 40 primes along one diagonal. Interestingly, of the first 2,398 numbers generated by the formula, precisely half are primes. Checking all such numbers less than 10 million, Ulam and his coworkers found the proportion of primes to be 0.475.
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The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists. The prime numbers are distributed among the natural numbers in a (so far) unpredictable way, but there do appear to be laws governing their behavior. Leonhard Euler commented
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[One] interesting fact about prime numbers is the formula developed by Leonhard Euler that produces a list of prime numbers. This formula only works for n = 0 to n = 15.
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Leonhard Euler