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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Formula for Primes?

Is there a nice cozy formula that will always spit out primes? Try this one:

f(n) = n2 + n + 41.

Euler discovered that this formula has a long string of prime values: it is prime for all n between 0 and 39 inclusive. However, it is not prime for all integers. In fact, it can be shown that no non-constant polynomial with integral coefficients will always spit out primes at the natural numbers.

There are formulas which always spit out primes when you plug in a natural number... here's one (Mills, 1947):

greatest integer less than (X raised to 3n),

where X is approximately 1.3064... Surprised? See the remark below!

The Math Behind the Fact:
It is worth pointing out that while the formula above looks nice, it is useless... it grows too quickly, and to determine X is tantamount to knowing the primes in its range!

Su, Francis E., et al. "Formula for Primes?." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
P. Ribenboim, The Little Book of Big Primes

Keywords:    prime formula, number theory
Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
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