Tagged Questions

Questions on finding integer/rational solutions of equations.

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, \ ...
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 ...
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 ...
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 ...
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? ...
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 ...
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 ...
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 ...
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, ...
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 $n$?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 31, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9042055606842041, "perplexity_flag": "middle"}