Is it necessary for a number of the form $4k^2+1$ to have at least one prime factor of the form $4n+1$?

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Is it necessary for a number of the form $4k^2+1$ to have at least one prime factor of the form $4n+1$?

I was trying to prove that there are infinitely many primes of the form $4n+1$, but to prove it, I need to prove the above statement true.