A Diophantine Equation | m^2-16n^2=17

Показать описание

🤩 Hello everyone, I'm very excited to bring you a new channel (SyberMath Shorts)
Enjoy...and thank you for your support!!! 🧡🥰🎉🥳🧡

If you need to post a picture of your solution or idea:
#NumberTheoryProblems #DiophantineEquations
via @YouTube @Apple @Desmos @NotabilityApp @googledocs @canva

PLAYLISTS 🎵 :

SyberMath

Рекомендации по теме

Комментарии

Nice easy problem!

Here is my solution, where I only needed to consider a single case.

m²-16n²=17
⇔a²-16b²=17 where a=|m|, b=|n|.
(a+4b)(a-4b)=17
As a, b non-negative integers, a+4b>0 and a-4b>0 (as neither can be 0),
and as a+4b≥a-4b, we have
a+4b=17 and a-4b=1

Adding the equations, 2a=18, a=9.
Subtracting the equations, 8b=16, b=2.
So |m|=9, m=±9
and |n|=2, n=±2

So we get four solutions for (m, n):
(9, ±2) o(-9, ±2).

MichaelRothwell

(m + 4n)(m - 4n) = 17
case 1:
m + 4n = 17,
m - 4n = 1
m = 9, n = 2
case 2:
m + 4n = 1,
m - 4n = 17
m = 9, n = -2
case 3:
m + 4n = -17
m - 4n = -1
m = -9, n = -2
case 4:
m + 4n = -1
m - 4n = -17
m = -9, n = 2
{(m, n)} = {(9, 2), (9, -2), (-9, -2), (-9, 2)}

rob

Thank you for your nice serial numbers

ezzatabdo

(m, n)=(9, 2) or (-9, -2) or (9, -2) or (-9, 2)

mathswan

Actually you could have stopped your solution when you got a solution (9, 2)
By seeing the question as it is square you can get the other values easily no need to solve more
So (9, 2), (-9, 2), (9, -2), (-9, -2)

msathwik

Isn't m - 4n < m + 4n? If so, that means 2 of the solutions may be invalid

robertsandy

m^2 - 16n^2= 17
The third binomic formula yields
(m + 4n) * (m - 4n) = 17
Since m and n are integer numbers, m + 4n and m - 4n are integers as well.

The only possible integer factorizations of 17 are
1 * 17
17 * 1
(-1) * (-17)
(-17) * (-1)
The system
m + 4n = 1
m - 4n = 17
has the solution
2m = 18, thus m = 9
8n = -16, thus n = -2
The system
m + 4n = 17
m - 4n = 1
has the solution
2m = 18, thus m = 9
8n = 16, thus n = 2
The system
m + 4n = -1
m - 4n = -17
has the solution
2m = -18, thus m = -9
8n = 16, thus n = 2
The system
m + 4n = -17
m - 4n = -1
has the solution
2m = -18, thus m = -9
8n = -16, thus n = -2
Altogether, we have four solutions:
{(+9, +2), (+9, -2), (-9, +2), (-9, -2)}.
which are symmetrisch to zero since it m^2 and n^2 take the same values for +-9 and +-2.

goldfing

Wow one I can solve! Since diff of 2 squares, and 17 factors only to 1*17, (m, n) = (+-9, +-2).

Qermaq

Diff of squares - always comes up in number theory type of problems. Just don't forget the negatives!!?

mcwulf

m=|9|, n=|2|
17+144=161
17+256=273
17+400=417
17+576=593
17+784=801
17+1024=1041
17+1306=1323
17+1600=1617
.
.
.

rakenzarnsworld

A Diophantine Equation | m^2-16n^2=17

A Diophantine Equation | m^2-16n^2=17

an exponential Diophantine equation.

A Linear Diophantine Equation | 3x+4y=17

A simple Factorial Diophantine equation (17x^2=9y!+2003)

A Diophantine Equation | Integer Solutions

Diophantine Equation | Integer Solutions?

A Nice Diophantine Equation | Integer Solutions

A Diophantine Equation | m=n/(n+5)

what a nice Diophantine equation.

All Solutions to the Diophantine Equation a^2+b^2=2009 | Euclidean Division | math Olympiad

A Nice Diophantine Equation | Math Olympiad Preparation | Challenging Algebraic Problem.

A Diophantine Equation | x^y=y^x

Solving A Homemade Diophantine Equation

Solve 2^x + 17 = y^4 | Polish Mathematical Olympiad 2014

Solving m^2=n^3+k^2 (partially) for Positive Integers

Quadratic Equations IIT Questions No 11 ( X Class)

Amazing Diophantine Equation

Module 2:- Diophantine Equations and Primes

A Nice Exponential Equation from Romania

Linear Diophantine Equation |Examples |Number Theory

A Nice Diophantine Equation in Number Theory | You Should Learn This Theorem | Math Olympiad

Math 361 - Diophantine Equations

I Solved A Quadratic Diophantine Equation | Integer Solutions?

RMO 2017 Problem 2 - Diophantine modulo 7