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Show that the polynomial function $$f(m,n)=(m+n−2)(m+n−1)/2+m $$ is one-to-one and onto. Both domain is $\Bbb Z^+\times \Bbb Z^+$, codomain are $\Bbb Z^+$.

I want to prove $f(a,b)=f(c,d) \longrightarrow (a=c \text{ and }b=d)$. In the end,I got

$$a^{2} + b^{2}+2ab-a-3b=c^{2} + d^{2}+2cd-c-3d.$$

What should I do then? Lets $a>c$ but I have no clues.

Thanks

Teddy38
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Moly Holy
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  • This is neither one-to-one nor onto. –  Jun 18 '20 at 11:36
  • An enarged question can be found there :https://math.stackexchange.com/questions/2707939/bijective-polynomials – Jean Marie Jun 18 '20 at 11:43
  • If you write a short program to calculate the values for some test values of $m$ and $n$ you get this: $$ 1\ 2\ 4\ 7\ 11\ 16\ 22\ 29\ 37\ 46 \ 3\ 5\ 8\ 12\ 17\ 23\ 30\ 38\ 47\ 57 \ 6\ 9\ 13\ 18\ 24\ 31\ 39\ 48\ 58\ 69 \ 10\ 14\ 19\ 25\ 32\ 40\ 49\ 59\ 70\ 82 \ 15\ 20\ 26\ 33\ 41\ 50\ 60\ 71\ 83\ 96 \ 21\ 27\ 34\ 42\ 51\ 61\ 72\ 84\ 97\ 111 \ 28\ 35\ 43\ 52\ 62\ 73\ 85\ 98\ 112\ 127 \ 36\ 44\ 53\ 63\ 74\ 86\ 99\ 113\ 128\ 144 $$ Should be enough for a hint. – Teddy38 Jun 18 '20 at 11:48
  • @ProfessorVector The domain is positive integers. – Teddy38 Jun 18 '20 at 11:48
  • Correct the title of your question, please, it is inconsistent with the body. –  Jun 18 '20 at 11:55
  • @bof i editd it and i found useful resources https://math.stackexchange.com/a/91323/620871 can some explain n−q=ad+(d(d+1))/2≥a+1? Thanks – Moly Holy Jun 18 '20 at 13:44

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