# What is the largest prime less than 2^31?

I'm sorry for this kind of specific question, I'd love if you could link to resources (prime lists, etc) that can answer similar questions more generically.

• Mathematica should be able to answer this question quickly; it has a function that will tell you how many primes there are less than 2^{31} and another that tells you what the nth prime is. Use one, then the other. – Qiaochu Yuan Nov 12 '10 at 22:20
• These are all great answers. Thank you everyone. – Martin Nov 12 '10 at 22:38
• @Qiaochu: A shortcut is NextPrime[2^31,-1]. – Hans Lundmark Nov 12 '10 at 23:31
• ...and it works on Wolfram Alpha too: wolframalpha.com/input/?i=NextPrime%5B2%5E31%2C-1%5D – Hans Lundmark Nov 12 '10 at 23:33

http://www.prime-numbers.org/prime-number-2147480000-2147485000.htm tells you that it's 2147483647 (about 2/3rds of the way down, third column). This website seems like a good resource if you're looking for lots of primes.

• Thank you, that list was exactly what I needed. – Martin Nov 12 '10 at 22:39

It is $2^{31}-1$. You might want to check Mersenne prime for similar details.

http://en.wikipedia.org/wiki/Mersenne_prime

• Interestingly this is one of the four known Mersenne double prime.en.wikipedia.org/wiki/Double_Mersenne_number – user17762 Nov 12 '10 at 22:20
• The fact that this is a prime is taken advantage by pseudo random number generators on $32$ bit machines. – user17762 Nov 12 '10 at 22:22
• The first proof of primality was given by Euler, and it remained the largest-known prime for nearly 100 years. – Douglas S. Stones Nov 12 '10 at 22:30
• Do you have a link to the proof @douglas? – AnonymousCoward Nov 12 '10 at 22:39
• Here's Euler's proof: math.dartmouth.edu/~euler/pages/E461.html Although you might be more interested in the wikipedia page: en.wikipedia.org/wiki/2147483647 – Douglas S. Stones Nov 12 '10 at 23:07