Useful restrictions for prime factors of a sum of two powers with coprime exponents

Every prime factor of a number of the form

$$a^2+b^2$$

with gcd(a,b)=1 has -1 as a quadratic residue.

Does this work only for exponents with a common factor, or are there useful restrictions also for coprime exponents ?

For example , are there any restrictions for the prime factors of numbers of the form

$$a^4 + b^9$$

with gcd(a,b)=1 ?

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Consider $a^m+b^n$. Since you want the exponents to be coprime, at least one of them (say $n$) is odd. Given a prime $p$, just take $a\equiv1$ and $b\equiv-1\pmod p$, then $p\mid a^m+b^n$. You might want to impose more restrictions on the exponents. –  barto Jan 21 at 11:35