What numbers can be expressed with the following expression?

Given $\displaystyle \frac{a^3 - b^3}{c^3 - d^3}$, where a,b,c and d are distinct prime numbers, which integers can be expressed?

Somebody asked this elsewhere online and it is beyond my abilities. I tried for quite a while but can't seem to form a good proof for the answer. The original poster initially asked "which positive integers can be expressed?" I used "numbers" because I have not yet determined if only positive integers can be expressed with this. Please explain your reasoning since I want to become stronger in this type of number theory. If this has been covered on this site, you may just link that existing answer. I tried to find it but was unable to. Thank you!

My current thinking:

In simple terms (since I'm not gifted in proofs), my working hypothesis is that I can make the difference of the numerator very large if I wanted to. Second, by selecting a>b or b>a, I can choose the sign of the result. This therefore leads me to believe I can express very large negative and positive integers. Now, I believe it is also possible to have the division of the numerator and denominator result in a non-integer value. This seems to imply any rational number can be expressed with this expression. However, I am not confident that my selection of primes (limited to all in existence) is able to express every possible decimal value to arbitrary precision and this leads me to believe I may have the same restriction with some integers, namely, gaps in the integers that can be expressed. I know there are an infinite number of primes, but not sure if that means I can always find a combination of them that would result in any integer or decimal value. Am I even close? Getting warm? Also, in your solution if you want to pepper in some great prime properties that would be great too. I know the fundamental theorem of arithmetic and have read about Goldbach's conjecture, but I suspect there are a few other properties that make this question immediately obvious.

Here is the original question: http://answers.yahoo.com/question/index?qid=20140225055002AARvwEX

I hope this is an unanswerable question in its current form. This would be better for my self esteem!

migrated from mathoverflow.netMar 2 '14 at 8:21

This question came from our site for professional mathematicians.

• I doubt that very much can be said about these numbers. "Somebody asked this elsewhere online" --- can you tell us where? – Gerry Myerson Mar 2 '14 at 11:53
• Maybe a concrete question can be asked. What is the least positive integer that cannot be written in this form? – GEdgar Mar 2 '14 at 13:20
• @GEdgar Even more concrete: can we express $1$? – Jack M Mar 2 '14 at 18:35
• @Jack M I thought the same thing. Can we prove that the difference between the cubes of two primes can be equal to the difference of two other cubed primes? That may be the key to this.So far, that proof is beyond me. – Trekgeek1 Mar 2 '14 at 18:39
• That's asking whether there's a number that's a sum of two cubes of primes in two different ways. – Gerry Myerson Mar 3 '14 at 8:38

Numbers expressible as $p^3 + q^3$ with $p$, $q$ prime in at least two ways at the Online Encyclopedia of Integer Sequences has many examples, the smallest being $$6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3$$ So this means $${1823^3-1699^3\over1049^3-61^3}=1$$ This at any rate answers a question raised in the comments.