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This may be a dumb question, but it bothers for quite a while.

Lets say, we have a certain equation, like $ab-a$ where $a, b$ are primes. Then we generate a sequence for every $a$ and $b$ which looks like $2,3,4,5,6,7,8,10,11,12,13,14,17,...$

Is this possible to prove, that there exists a finite number of equations that use only two variables $a$ and $b$ (which are both prime) that altogether generate a set of natural numbers? Is this problem (or similar) relevant to something?

Are
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  • Do you mean "generate a set of prime numbers"? – M10687 Jan 06 '16 at 01:30
  • $a+b$,$a+b-1$,$a-a+2$,$a-a+1$ might generate all positive numbers. – Empy2 Jan 06 '16 at 01:32
  • @M10687 No, what I meant is that the equation that I've shown above generates a set which looks like a set of natural numbers with "holes" in it. So I was wondering, if there exists a certain number of equations that altogether generates a complete set of natural numbers. – Are Jan 06 '16 at 01:33
  • Oh I see what you're saying now, sorry about that. – M10687 Jan 06 '16 at 01:35
  • @Michael How can you be certain? Can you prove it? Or explain to me how did you get those equations? – Are Jan 06 '16 at 01:37
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    They are Goldbach's conjecture, that every even number greater than 2 is the sum of two primes. – Empy2 Jan 06 '16 at 01:42
  • It's worth noting that $a+b+c+d$ works due to recent work. Two variables is harder. Also, depending on what sort of expressions you mean, you can do it in one variable - for instance $\lfloor \log_2(a)\rfloor$ includes all non-negative integers (most of them multiple times, even!). If you just meant polynomials or algebraic functions that's harder (and I'll bet that only linear functions are particularly relevant - I would think higher order polynomials would grow too fast and cover too sparse a set to contribute anything) – Milo Brandt Jan 06 '16 at 01:54

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Absolutely.

Goldbach's conjecture tries to cover positive even integers' space with f(a,b)=a+b

Obviously if the conjecture holds you'd just add f(a,b)=a+b+1 and so on.

Welcome to number theory!

numbquery
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