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$x^6 y + x^6 - x^5 y^2 + x^4 y^2 + x^4 y - x^3 y^4 - x^3 y^3 + x^3 y^2 + x^2 y^5 + x^2 y^3 - x y^5 = z, x > y$

... over the natural numbers?

Just curious what you call this kind of polynomial equation, as I am a non-mathematician doing some learning and research outside of any kind of school.

CommaToast
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  • First and foremost, this is a polynomial of two variables, in this case of degree 7. – blub Aug 11 '18 at 03:10
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    What is an idoneal? – Allawonder Aug 11 '18 at 03:45
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    It's not an equation: an equation has an equals sign. – Angina Seng Aug 11 '18 at 03:46
  • For idoneal numbers see this question. How did you derive this polynomial from "an idoneal number"? – Dietrich Burde Aug 11 '18 at 15:38
  • @DietrichBurde Not enuf space to explain it here. I am trying to teach myself some combinatorics and number theory etc. If you really wanna know I could write it up as an arxiv.org thing if you wanna see it. But warning: I am a total amateur armchair math non-genius... lol – CommaToast Aug 18 '18 at 06:59
  • Ok I revised the question, this should make it clearer what I am asking. – CommaToast Aug 21 '18 at 12:17
  • It is still not clear to me what you are asking. Do you want to know how are called the polynomials which generate only natural numbers when they are given natural numbers, or do you want to know something about idoneal numbers? – Jose Brox Aug 21 '18 at 12:22
  • Forget about idoneal, I realized its irrelevant to this question. Just wanna know what to call a polynomial equation which takes in and gives out only natural numbers. I am using it to explore combinatoric number theory, en route to more complex things. – CommaToast Aug 21 '18 at 12:28

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This is just the expansion of $$-a \left(a^2-a b+b\right) \left(b^2-a b-a\right) \left(a^2+a b+b^2\right)$$

Claude Leibovici
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  • That is... similar to what I expanded it from, though not identical. I guess I was more wondering if this is a special flavor of polynomial which has a name that I can google to learn more about this. I am trying to teach myself combinatorics and some number theory, and it’s kinda slow going. – CommaToast Aug 18 '18 at 07:01
  • I revised the question, have another look? – CommaToast Aug 21 '18 at 12:17