# Are there any primes of the form 38^N+31?

Asked by

PhiNotPi (

12609)
March 17th, 2014

This is a math question I found on another site (source), and it has been bugging me because I haven’t been able to determine the correct answer.

Simply put, is there a prime of the following form? If so, what is that prime?

p = 38^N + 31

(Where N is an integer.)

Computer calculations have not been able to solve this problem yet.

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## 12 Answers

The problem with question alike this is you can keep calculating ‘til the cows come home from jumping over the moon, but you’re never going to know for sure until you find a prime that fits the equation.

Just out of curiosity, what is special about 38 and 31? Have there been investigations into the prmality of a^n + b for other values of a and b? The only one that I know about are the Mersenne primes 2^n – 1, which some people are rather obsessed over

Does that formula intersect with the Marsenne prime formula? (Looks like no to me, but I’m not sure.)

@LostInParadise Nothing is particularly special about these two numbers, other than the difficulty of finding a solution. Most combinations of numbers have very simple solutions.

@bolwerk No, this does not seem to intersect with the Mersenne prime, but the fact that the formulas are similar makes me think that similar solution methods may work on both.

If N is Zero then p might be 1 and 31 might be a prime

@PhiNotPi Sorry… I’m new to university algerbra… I am used to the easy high school answers. Are you allowed to use imaginary numbers?

@talljasperman No, you aren’t allowed to use imaginary numbers in this case.

Also, I’ve been following the original thread. They have not been able to find a prime yet, but they know that the value of N is most likely larger than 200,000.

@PhiNotPi Would this be the time to use a quantum computer? Maybe you can’t find an answer because the answer is no.

@talljasperman The answer probably is “no” at this point. It is theoretically possible to prove that the answer is no, but for some reason that is being very hard to do in this case.

Just to update the progress, it is believed (by the users on the other website) that a solution *does exist.* The problem is that the expected solution is very large, with N > 1 million. It is currently impossible to test such large numbers on ordinary computers.

A teacher I had made an interesting statement about the difficulty of proofs. He said that in general the simpler the numbers you work with, the more difficult the proof. Proofs involving complex numbers are easier than ones involving real numbers, which are easier than proofs involving whole numbers. For example, Fermat’s equation is easily stated, but the proof involves some very advanced math that is way beyond what I know.

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