Pythagorean triples that “survive” Euler’s totient function

Suppose you have three positive integers a,b,c that form a Pythagorean triple:
a2+b2=c2.
Additionally, suppose that when you apply Euler’s totient function to each term, the equation still holds:
ϕ(a2)+ϕ(b2)=ϕ(c2).
One way this can happen is if a2,b2,c2 have the same primes in their prime factorization. (For example, starting from the Pythagorean triple 3,4,5, we could multiply all three terms by 30 to get 90,120,150. If we do, then we have 902+1202=1502 and ϕ(902)+ϕ(1202)=ϕ(1502).) In that case, because all three terms are squares, they all contain these prime factors at least twice, and so we must have
ϕ(ϕ(a2))+ϕ(ϕ(b2))=ϕ(ϕ(c2)).
My question is: are there any “atypical” solutions to the two equations (1) and (2) for which (3) does not hold? Or at least where (1) and (2) hold, but the prime factorizations of a,b,c do not consist of the same primes, even if (3) happens to hold for a different reason?


In the comments, Peter and Gerry Myerson have checked small cases (all triples for 1ab105 and primitive triples generated by (m,n) for 1nm2000) without finding any atypical solutions.

Here is an in-depth explanation for why typical solutions like (90,120,150) work. By a typical solution, I mean a solution where a,b,c have the same primes in their prime factorization. Such a triple satisfies (2) and (3) whenever it satisfies (1), as shown below.

Let rad(x) denote the radical of x: the product of all distinct prime factors of x. To get a typical solution, we start with any Pythagorean triple, then scale (a,b,c) so that rad(a)=rad(b)=rad(c)=r.

It is a general totient function identity that whenever rad(x)=r, ϕ(x)=ϕ(r)rx. In other words, ϕ(x)=xpxp1p where the product is over all primes p that divide x.

In the case above, we have
ϕ(a2)+ϕ(b2)=ϕ(r)ra2+ϕ(r)rb2=ϕ(r)rc2=ϕ(c2),
and (2) holds.
Moreover, since ra,b,c, we have r2a2,b2,c2, so when we multiply by ϕ(r)r, we have rϕ(r)ϕ(a2),ϕ(b2),ϕ(c2). Therefore all prime factors of rϕ(r) divide each of ϕ(a2), ϕ(b2), and ϕ(c2). These are all their prime factors, since r contained all the prime factors of a2,b2,c2 and since then the only new prime factors introduced came from multiplying by ϕ(r).

As a result, ϕ(a2),ϕ(b2),ϕ(c2) still have the same set of prime factors: rad(ϕ(a2))=rad(rϕ(r))=s, and similarly rad(ϕ(b2))=rad(ϕ(c2))=s. So (3) holds, because
ϕ(ϕ(a2))+ϕ(ϕ(b2))=ϕ(s)sϕ(a2)+ϕ(s)sϕ(b2)=ϕ(s)sϕ(c2)=ϕ(ϕ(c2)).

Answer

Possible help:

Every Pythagorean triple is of form (2mn,m2n2,m2+n2)
If m, n are not coprime then these all share a factor. But it can be proven that all pythagorean triples have:

  • at least 1 entry divisible by 3
  • at least 1 entry divisible by 4
  • at least 1 entry divisible by 5

not necessarily all distinct, As Euler’s phi function is multiplicative these may come into play.

A proof follows:

If at least one of m, n is divisible by 3, it follows that all of the values are divisible by 3. If neither is, it follows that their squares are both remainder 1 on division by 3 so m2n2 will divide by 3. If at least 1 of m, n are even 2mn is divisible by 4. Otherwise, both have squares that have remainder 1 on division by 4, so m2n2 is divisible by 4. Lastly, if at least 1 of m, n is divisible by 5, then 2mn is divisible by 5. If their squares have the same remainder on division by 5, then m2n2 is divisible by 5. Finally, if the remainders of the squares are different then m2+n2 is divisible by 5.

Since Euler’s phi function is multiplicative, multiplying it’s outputs for any pair of coprime values as inputs, will create the output for their product. Any case where the 4 and 5 don’t overlap forces the other leg to follow suit and give a Euler phi output it’s a multiple of 4. Since any multiple of 4 except 4 itself, has a output that is also a multiple of 4, as will any product with 2 or more odd primes. It seems nearly certain parity won’t be broken by equation 3 in any case where equation 2 holds.

Attribution
Source : Link , Question Author : Misha Lavrov , Answer Author : Xander Henderson

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