4
votes
4answers
897 views

Is there an analogue of finite fields for products of two prime powers?

The collection of prime powers can be characterized in the following way: There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...
0
votes
1answer
386 views

Primes are to Irreducible Polynomials as Prime-related theorems are to ?? [closed]

Irreducible polynomials are often introduced as the analog to prime numbers in polynomial rings. Prime numbers, of course, have a very rich theory, leading to the likes of the Riemann Zeta function ...
9
votes
2answers
845 views

Non-Standard Prime

Hello, My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$: $c \neq 0$, $c \neq 1$, $c \neq ...
28
votes
3answers
2k views

What is the current status of Agrawal's conjecture?

In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture: If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in ...
2
votes
1answer
994 views

Is there an algebraic proof of the infinitude of primes? [closed]

It is well-known that there exists a (justly celebrated) topological proof of the infinitude of primes (Hillel F├╝rstenburg, 1955). Does there also exist an algebraic proof?
9
votes
2answers
1k views

Gaussian primes, quaternion primes, … octonions?

Is there a notion of an octonion prime? A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime. A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is ...