Mike Bennett
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 Apr 27 comment How many pairs $(a,b)$ of integers such that , $a^2b^2=4a^5+b^3$ There are precisely $8$ solutions. I won't post more until Gerry's question is answered. Apr 22 answered $2^n + 3^n = x^p$ has no solutions over the natural numbers Apr 22 comment $2^n + 3^n = x^p$ has no solutions over the natural numbers An old version of this paper is available on the arxiv : arxiv.org/pdf/1603.07922.pdf Apr 21 comment $2^n + 3^n = x^p$ has no solutions over the natural numbers This equation has no solutions. The proof of this fact, however, is not easy, as far as I know (it appears as a special case of a somewhat more general result in a paper of mine with Nicolas Billerey to appear in Math. Comp.). Our proof relies upon Frey curves. Apr 20 comment How does zeta of zero equal to negative one half rather than to infinity? The answer to your first question is "no". Read about analytic continuation. Apr 8 comment If $11^m\cdot 5^n-3^p\cdot 2^q=1$ where $m,n,p,q$ are non-negative integers,Find all $m,n,p,q$ You can do this working modulo $2^2 \cdot 3 \cdot 5^2 \cdot 11$ and invoking Mihailescu (or some weaker version). Apr 4 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. I give up. You appear to be reasonably determined not to understand. Apr 4 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. No, I meant odd. There certainly can exist such primes dividing $x^4-4$ an even number of times (such as when $x=4$). That's not the point. Apr 4 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. So, does there exist a prime $p \equiv 3 \mod{4}$ that divides $x^4-4$ an odd number of times? Apr 4 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. If $x$ is odd, what common factors can $x^2-2$ and $x^2+2$ have? Apr 4 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. Now answer the same question about $x^2-2$. Apr 3 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. The point is that, yes, every prime $p \equiv 3 \mod{4}$ divides a sum of two squares an even number of times. Apr 1 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. So if $p \equiv 3 \mod{4}$ is prime, how many times can $p$ divide the right hand side? Apr 1 comment Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. Factor $x^4-4$ and use what you know about sums of squares. Mar 29 answered Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$. Mar 18 comment Seeking general methods to attack $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4=z^2$ in integers/rationals Yes, but you need to work out multiples of the generator on the elliptic curve (these don't correspond to multiples of the solutions (x,y,z)). Mar 17 comment Seeking general methods to attack $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4=z^2$ in integers/rationals Actually, I suspect that this would be very hard to show (or even not true -- one could search for counterexamples easily enough). Any solution with $y$ prime would lead to an integer, for example... Mar 17 answered Seeking general methods to attack $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4=z^2$ in integers/rationals Mar 16 comment Is it possible to show that there are integer solutions $n,m$ for $10^m+10^n+1\equiv 0$ (mod $q$) for a prime $q$? You might want to take a look at $q=11$. Jan 23 awarded Yearling