# Tagged Questions

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### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...
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### $p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
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### The equation $x^m-1=y^n+y^{n-1}+…+1$ in prime powers $x,y$

Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$? This question arose in the ...
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### The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
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### Catalan-type equations for prime powers

Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers? For example, if there are infinitely many Sophie ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X^2-PY^2=k$ ...