Question

prove/disprove
a prime number p can't be written as sum of 3 squares .

Answer 1

\(\large\tt \begin{align} \color{black}{
6^2+6^2+1^2=73


}\end{align}\)

Answer 2

hmm counter example :|

Answer 3

that works actully

Answer 4

simplest example


\(\large\tt \begin{align} \color{black}{
1^2+1^2+1^2=3


}\end{align}\)

Answer 5

you want to find a prime that cannot be written as sum of squares of 3 integers ?

Answer 6

i just wanted to be confident as much as i can about primes properties hmm
at which type of primes we can write them as sum of 3 squares :|
idk if its even possible

i wanna any integers

Answer 7

you cannot write primes of form 8k-1 as sum of squares of 3 integers

for example : 23 cannot be written as sum of squares of 3 integers

Answer 8

what about others?

Answer 9

like what about
8k+7

Answer 10

thats same as 8k-1

Answer 11

lol ithought that was 8k+1 :(

Answer 12

in mod8 primes can only be of form
8k+1
8k+3
8k+5
8k+7

Answer 13

you know that primes of form 8k+7 cannot be expressed as sum of 3 squares

you want to analyze the remaining ?

Answer 14

ok i got now why i cant

Answer 15

yes sure

Answer 16

why ?

Answer 17

8k+1=2(2k+1)^2-(8k^2+1) ?

Answer 18

so ?

Answer 19

xD
ok show me what got

Answer 20

Notice a square only leaves remainders : {0, 1, 4} in mod8

Answer 21

yes

Answer 22

taking any combination of them can never add up to 7 \(\blacksquare \)

Answer 23

i see

Answer 24

0 + 0 + 0 = 0 mod 8
1 + 1 + 1 = 3 mod 8
4 + 4 + 4 = 4 mod 8

0 + 1 + 1 = 2 mod 8
0 + 1 + 4 = 5 mod 8
...

Answer 25

it is never 7 mod 8

Answer 26

infact that proves that no integer of form 8k+7 can be expressed as sum of 3 squares

see this
http://math.stackexchange.com/questions/779784/show-that-an-integer-of-the-form-8k-7-cannot-be-written-as-the-sum-of-three

Answer 27

ok got it