Integer Circles

Number Theory Level 3

Given the equation of a circle as ${ x }^{ 2 }+{ y }^{ 2 }=1234567$ .How many points on the circle have integer coordinates?

This problem is a part of my set The Best of Me

The answer is 0.

2 solutions

Siddharth G
Aug 15, 2014

Taking modulus 4 , we get
${ x }^{ 2 }+{ y }^{ 2 }\equiv 1234567(mod\quad 4)\\ ({ 0 }^{ 2 },{ 1 }^{ 2 },{ 2 }^{ 2 },{ 3 }^{ 2 })+({ 0 }^{ 2 },{ 1 }^{ 2 },{ 2 }^{ 2 },{ 3 }^{ 2 })\equiv 3(mod\quad 4)\\ (0,1,4,9)+(0,1,4,9)\equiv 3(mod\quad 4)\\ (0,1,0,1)+(0,1,0,1)\equiv 3(mod\quad 4)\\ (0,1)+(0,1)\equiv 3(mod\quad 4)\\ 0,1, 2\equiv 3(mod\quad 4)\quad$
Hence there are 0 solutions.

why modulu 4?

Ori Ashual - 6 years, 8 months ago

Modulo 3,4 are standard while dealing with squares.

Siddharth G - 6 years, 4 months ago

Highly interesting question. Quite did the same until the co-ordinate geometry put me off :).

Krishna Ar - 6 years, 9 months ago

can't you just see whether 1234567 is a pythagorean triplet or not? wouldn't that be enough? if it's not, then, no integer value for x and y.

Sarthak Rath - 6 years, 3 months ago

How would we figure out whether 12345657 is a part of a pythagorean triplet or not?

Siddharth G - 6 years, 3 months ago

Arijit ghosh Dastidar
Mar 2, 2015

I used Fermats two squares theorem too..

Integer Circles

The answer is 0.

2 solutions

0 pending reports