Questions on finding integer/rational solutions of equations.
7
votes
3answers
632 views
Proving a statement regarding a Diophantine equation
FINAL EDIT : Prove that if $p^z|n^2-1$
$$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$.
Here $p>3$ is an odd prime , $x=2y+z, \ ...
11
votes
2answers
206 views
How to find a “better description” (e.g. recurrence relation) for this sequence?
My solution to a problem in Project Euler required to solve this subproblem: find values of $k\in\mathrm{N}$ such that $3k^2+4$ is a perfect square.
As I was writting a computer program, I just tried ...
-1
votes
2answers
56 views
Context problems of Number theory and functional equation
I can't solve the following problems, please help.
1) Find all primes $p$ and $q$ such that $p^q+q^p$ is a prime.
2) Solve $2^x+3^y=z^2$ in integers.
3) Find all $f: \mathbb{Q} \rightarrow ...
3
votes
1answer
458 views
Integer coordinate set of points that is a member of sphere surface
I have a graphic application to develop which involve many spheres. I should determine then on run time.
Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
0
votes
1answer
40 views
find all positive integers for a given diophantine equation involving 4 or 7 variables
Given equation:
Ap + Bq + Cr + Ds + Et + Gu + Vg = K; (Eq. in 7 variables);
suppose we have A, B, C, D initialize with = 1,2,5,10,20,50 and 100 respectively; and K = 50000;
How do we solve it?
...
0
votes
1answer
79 views
Nice sequences related to the Diophantine equation $d^{m+1} =a^{m}+ b^{m}+ c^{m}$
$$1, 3, 12, 32,...$$
Above is the sequence of the number of solutions, if there are, to the Diophantine equation :
$d^{m+1} =a^{m}+ b^{m}+ c^{m}$ for $m =2$, in positive integers where $a, b$ and ...
0
votes
0answers
91 views
another intresting equation $p^x$ + $q^y$ -$r^z$ = 0
members, the equation $p^x$ + $q^y$ - $r^z$ = 0 (where r is odd integer) has only positive integr solutions iff the following conditions made.
a) p = -1 (mod q^(2k)), here k is any positive ...
0
votes
0answers
168 views
Finding solutions of the inequality
I have seen the lecture note of Prof. Gandhi from BITS, and I could not digest to obtain (X, Y, Z) values in integers. Is there a method to analyze the nature of solutions in integers of the following ...
0
votes
0answers
472 views
Using recurrences to solve $3a^2=2b^2+1$
Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, ...
0
votes
0answers
114 views
Diophantine equation system containing modulo
Is there any method to find all integers $(x,y,n)$ satisfying
$8346192 = (1193363x+y)$ mod $n, $
$6550593 = (8346192x+y)$ mod $n, $
$3632765 = (6550593x+y)$ mod $n$?
