1) Very few mathematicians work with large numbers or indeed with any numbers at all. 2) Did you just rediscover well-known Chinese Remainder Theorem tricks?

I do not understand what you mean by "can't get into contact..." Google university math depts to get email addresses, and send email giving some details...

It is not the job of mathematicians in university departments to check whether software that random people send them works. How have you tried to contact mathematicians? How have you chosen who specifically to contact, and what did you ask them for?

Look for experts in “computational number theory”. Those are the people who may be interested in your idea, if anyone would be. And give them a reason to want to engage with you. At least from your post here one doesn’t get a strong sense that you know what you’re talking about. An email from a secretive amateur mathematician who wants my help, is cagy about disclosing any details, doesn’t offer to pay, and yet apparently believes there are large amounts of money to be made from “storing large primes”, would look to me quite hard to distinguish from spam.

Could you clarify in which way you plan to earn money by means of your program? It seems unlikely that university math departments will be willing to spend significant funds to buy software that does computations with large primes (even less so if the software first needs to be checked and the required hardware is not set up, yet).

Magma computed 2^80000000 in a few seconds. What actually do you think your software does that Magma (for example) doesn't do? And if you are cagey with the details no mathematicians will be interested, because you cannot prove your work is correct. You won't make money with this by going to maths departments, I bet.

Can you find a way to demonstrate that you have accomplished something interesting without disclosing how you did accomplish it?

You say that you are trying to get in contact with mathematicians so as to tailor the software to their needs. Have you considered that you might not be receiving replies because no one you've contacted needs your software?

How do you envision making money from this? From selling software? From providing a service? From a patent?

Space in the cloud?

"It would cost a pittance for universities to front the hardware that is invaluable to me" - I cannot see you making any progress with this attitude, which seems incredibly entitled and disconnected from understanding anything about the mission of universities.

I think you're sadly mistaken about the commercial prospects for computational number theory (outside of cryptography). There is no money in these problems. Mathematicians work on them as a form of art for art's sake. If you had a machine that applied paint to canvases in a new way, do you really expect artists would line up to buy your product? Print shops might, but artists?

Can you use your idea to mine cryptocurrency more efficiently than your competitors? That would solve the funding problem to some extent.

Why would a university front hardware for a company?

You have made a seriously profitable mathematical; discovery, yet you don't know any mathematicians personally? Your first step should be to find a mathematician that you know personally and get them to check your work (after signing an NDA, and for a small fee). Anyone with a PHD will do. If they think it;s feasible I'm sure they will introduce you to someone in the right field who you can pay to do a thorough evaluation

@DJClayworth: Nobody competent is going to sign an NDA to do work with a small fee. It's a huge risk to one's future livelihood (what the the NDA-covered material somehow intersects with something you're already working on or intending to work on?! or even if the other party merely falsely alleges or threatens to allege that it does... which seems very likely when dealing with crackpots). I would say for having a serious expert consider work involving an NDA in their field of expertise that's not scoped just to business matters (vs their domain)...

How much hard drive space do you really need? A standard 12 TB hard drive, which costs around $200, should be able to store a representation of a number around 2^(8*11*10^12), right? Isn't the largest prime computed so far around 2^(82,589,933)?

"My method is much more elegant." How do you know that? If you are a mathematician yourself, you don't need to ask this question. If you are not a mathematician yourself, what qualification you have to say that?

Hunters of new Mersenne prime usually do not need to "store very large prime numbers". Why do you need that for this? It seems likely that your technique is not useful. Of course, if you could locate a Mersenne prime before everybody else, that would be sensational. If you could even locate one with more than 100 million figures, that would earn you a prize. However, the discovery you (vaguely) describe does not seem to be useful for this.