# Tagged Questions

234 views

346 views

### Determining the exceptional set in the theorem of Ax & Kochen

Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...
658 views

### The “universal” diophantine equation

There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent All other diophantine equations (could be wrong on this) Any particular set ...
411 views

### Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
623 views

### Number Theory Representation of Primes

For a primes $p$ sufficiently large, does there always exists positive integers $k,a,b\in\mathbb{N}$ such that $p=(k+1)(ab)+k(a+b)$ or equivalently $p\equiv (ab)\bmod ((a+b)+ab)$? Please note that ...
### solution of the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$
i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.
### When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?
It is easy to show that the following problems are equivalent. a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square? and b. When is $X^2-PY^2=k$ ...