Final answer:
Odd pseudoprime numbers are composite numbers that pass tests for prime numbers, and there are infinitely many due to the existence of infinitely many Carmichael numbers, which are a special type of odd pseudoprime.
Step-by-step explanation:
Proof of Infinitely Many Odd Pseudoprimes
To prove that there are infinitely many odd pseudoprime numbers, we must understand what pseudoprimes are. A pseudoprime is a composite number that passes certain tests used to identify prime numbers, despite not being prime. Specifically, an odd pseudoprime is a composite number that is not divisible by 2 and satisfies the congruence bn-1 ≡ 1 (mod n) for some base b that is relatively prime to n. The most famous pseudoprime test is based on Fermat's Little Theorem.
Now, for odd pseudoprimes to be infinite, we can consider Carmichael numbers, which are a special type of pseudoprime that pass Fermat's test for all bases relatively prime to it. It has been proven that there are infinitely many Carmichael numbers, and thus, infinitely many odd pseudoprimes. For example, using the Chinese Remainder Theorem, one can construct Carmichael numbers by selecting suitable prime factors that meet certain criteria derived from Korselt's criterion, which states when a composite number is a Carmichael number. This yields an infinite sequence of non-prime odd numbers that are pseudoprimes.
Therefore, because Carmichael numbers are pseudoprimes and it has been shown that there are infinitely many of them, we can conclude that there are indeed infinitely many odd pseudoprime numbers.