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Suppose $a$ is a positive integer. When do the totient values of $a$ , $a+1$ and $a+2$ form a pythagorean triple ? In other words :

For which positive integers $a$ does the equation $$\phi(a)^2+\phi(a+1)^2=\phi(a+2)^2$$ hold ?

I yet only found the tripel $[80,60,100]$ appearing for $a=123$. Is it the only one ?

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  • $\begingroup$ A quick check for $a < 10^7$ shows only the trivial $a=0$ case and $a=123$. I would suspect that there are at most finitely many, but I'm not sure how I would try to prove it. $\endgroup$ – davidlowryduda Jun 21 '18 at 11:57
  • $\begingroup$ I ruled out $a=0$ and upto $a=10^8$, there is no further example. Currently, I am running the range $[10^8,10^9]$ $\endgroup$ – Peter Jun 21 '18 at 12:35

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