A Diophantine Equation | m^2-n^2=101

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every single odd number can be represented as the difference between two consecutive squares because the difference between two squares n and n+1 is (n+1)^2-n^2 = 2n+1. so 2n+1 = 101, 2n = 100, n = 50 and m = 50+1 =51. And then you can just flip through negatives to get the other solutions.

xchomphk.

Easy peasy - thank goodness for prime numbers!

scottleung

(m - n)(m + n) = 101, a prime. So so one of these four pairs of equations is satisfied:
m - n = 1 and m + n = 101
m - n = 101 and m + n = 1
m - n = -1 and m + n = -101
m + n = -1 and m - n = -101

There are two pairs of opposite-sign equations, meaning that m and n for those two are of opposite sign. Pick the top pair, and add the two equations to get 2m = 102 and m = 51. This makes n 50. The general case is m = ±51 and n = ±50. The signs need not match.

JohnRandomness

Since m²-n²=101
It implies m²>n² 👍
Great work 👍

tambuwalmathsclass

50;51 is the obvious
factor m^2-n^2 === (m+n)(m-n) & 101 is odd the trivial solution as above.
and singular as 101 is prime.

carlyet

m = 10
n = ✓-1
m² - n² = 10² - (✓-1)²
100-(-1) = 101
If One Of Them Can Be Made Complex

skyo

Isn't this a Hyperbola? I used Desmos
If x^2-y^2 =101 => (10.05, 0) & (-10.05, 0) are its vertices and same applies for y^2-x^2=101 but the points will be on y axis. Any of this makes sense related to your provided solution?

Shadi

Recommend a good book for number theory please as u said in the video

tanishkgoyal

m=51, n=50 (may be negative or positive)

rakenzarnsworld

Unless my memory is failing me, if a product of conjugates (a+b)(a-b) (which is what the difference of two squares translates to) equals a prime number, then their difference will be (a-b) = 1, and their sum (a+b) will equal the prime, and will be based on two numbers with a difference of 1. In this case, it's 51 and 50 without any brain work... Or 51^2 - 50^2 = 2601 - 2500 = 101. Perhaps you mention this in your video, but I didn't have 5 minutes to follow, sorry 🙂

SlawekM.

i first solved in my brain
m = 11
n = √20
but then i saw "integers"???
and i was like 😭💔

rythmx

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