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1d
comment Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $
In Pari/GP (which is freely available and documented) you can type : thue(thueinit(x^4-6*x^3-x^2+6*x-1),-1) to get a reply of [[-1, 1], [1, -1], [0, 1], [0, -1], [1, 1], [-1, -1], [15, 17], [-15, -17], [6, 1], [-6, -1]]. This "proves" that the solutions are as listed. The Thue solver here depends upon lower bounds for linear forms in (complex) logarithms and lattice basis reduction (facts which may or may not interest you).
Aug
23
comment Does $(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$ imply $(a,b,c)=(1,2,3)$?
I don't believe that the OP is looking for a polynomial identity (it is easy to show that there are no others). I suspect he (or perhaps she) simply wants to rule out the existence of quadruples $(a,b,c,n)$ for which $(\sum_{k=1}^n k^a)^b=\sum_{k=1}^n k^c$ with $(a,b,c) \neq (1,2,3)$, a rather subtler problem.
Aug
22
comment How to solve special type of Diophantine equation
The integer solutions to this equation have been determined by Bugeaud, Mignotte, Siksek, Stoll and Tengely : see arxiv.org/pdf/0801.4459v4.pdf
Jul
27
answered Generalization of Erdos-Selfridge
May
2
comment Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$?
There are no solutions modulo $24$.
Apr
25
comment Integers points of an elliptic curve
If you assume your curve to be given by a minimal model or some such (to avoid simply scaling rational points to give, supposing positive rank, as many integral points as desired), the current belief is that the number of "integral" points should be absolutely bounded.
Apr
21
awarded  Editor
Apr
21
comment Finding integer solutions of $m^2-n^5 = m - n$
There is no reason to believe that there is an elementary approach to solve this problem. Sometimes Diophantine equations are just hard.
Apr
21
revised Finding integer solutions of $m^2-n^5 = m - n$
added 1 character in body
Apr
21
answered Finding integer solutions of $m^2-n^5 = m - n$
Apr
15
answered Diophantine equation resembling FLT
Mar
3
answered Is $53$ expressible in this form?
Jan
23
awarded  Yearling
Dec
24
comment Does $p_{1}^x + p_{2}^y = n$ have uniqe solution for $x$ and $y$ ($p_{1}, p_{2}$ are primes).
Not to be unkind, but both the "proof' and the accepted answer here are incorrect.
Dec
19
answered Does $p_{1}^x + p_{2}^y = n$ have uniqe solution for $x$ and $y$ ($p_{1}, p_{2}$ are primes).
Dec
19
comment Does $p_{1}^x + p_{2}^y = n$ have uniqe solution for $x$ and $y$ ($p_{1}, p_{2}$ are primes).
How about $2^3+3=2+3^2$? The conclusion that $p \equiv 1 \mod{q}$ and $q \equiv 1 \mod{p}$ does not follow
Nov
29
answered Approximate irrational numbers with the same denominator
Nov
23
answered Solve $3^a-5^b=2$ for integers a and b.
Oct
4
awarded  Mortarboard
Sep
25
answered Is there any infinite set of primes for which membership can be decided quickly?