Odd primes congruent to 1 or 3 mod 4 are prime numbers that when divided by 4, leave a remainder of either 1 or 3. Examples include 5, 13, 17, 29, and so on.
Odd primes congruent to 1 or 3 mod 4 have important applications in number theory, specifically in the study of quadratic residues and the law of quadratic reciprocity. They also have practical uses in cryptography and coding theory.
The proof for this is based on the concept of modular arithmetic and the properties of prime numbers. It involves showing that if a prime number is not congruent to 1 or 3 mod 4, then it can be written as the sum of two perfect squares, which is a contradiction.
Yes, it has been proven that there are infinitely many odd primes congruent to 1 or 3 mod 4. This is known as Dirichlet's theorem on arithmetic progressions.
No, not all odd primes are congruent to 1 or 3 mod 4. There are prime numbers that are congruent to 2 or 0 mod 4. For example, 2 and 7 are primes that are congruent to 3 mod 4.