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!

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. 
