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Let S be the set of all numbers of the form $4n+1$ where $n \geq 1$. Prove or Disprove:

1) There are infinitely many S-primes

(I am thinking: if $p=4n+1$ is a normal prime then p is an S-prime, using some sort of a Euclidean type argument)

2) Every number in S can be written as a finite product of S-primes

(I am thinking an induction argument)

3) Every number in S can be written $\it{uniquely}$ as a finite product of S-primes

I know that a number is an S-prime if it is not divisible by any number in S other than itself. Thank you!

PrimeSpiralObserver
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2 Answers2

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The meaning of an $S$-prime is a number in $S$ that can not be written as a non-trivial product in $S$.

It is clear that a prime that is in $S$ is an $S$-prime, and there are infinitely many of these by a proof here.

Every odd prime is congruent to $\pm 1$ mod 4. The product of two primes each congruent to $-1$ mod 4 is an $S$-prime. Any element of $S$ is the product of some number of primes in $S$ and some even number of primes congruent to $-1$ mod 4. Hence it is the product of finitely many $S$-primes. But if it has more than 2 (i.e. $2k$ for $k >1$) prime factors congruent to $-1$ mod 4, then these can be grouped in pairs in various ways, so the $S$-prime factorization is not unique.

fred goodman
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For (3), consider $ 21 \cdot 21 = 9 \cdot 49$.

lhf
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