Prove-that-3k-2-is-not-perfect-square-for-all-k-0-1-2-3-

Question Number 23269 by Rasheed.Sindhi last updated on 28/Oct/17

P rove that

3 k + 2 is not perfect square for

all k \in {0, 1, 2, 3, \dots} .

Answered by Tinkutara last updated on 28/Oct/17

A n y n u m b e r c a n b e w r i t t e n a s 3 p,

3 p + 1 a n d 3 p + 2 .

I f i t i s 3 p, {i t}^{'} s s q u a r e = 9 p^{2} = 3 k f o r m

I f i t i s 3 p + 1, {i t}^{'} s s q u a r e = 9 p^{2} + 6 p + 1

= 3 k + 1 f o r m

I f i t i s 3 p + 2, {i t}^{'} s s q u a r e = 9 p^{2} + 12 p + 4

= 3 k + 1 f o r m

H e n c e a s q u a r e c a n n o t b e i n 3 k + 2

f o r m .

Commented by Rasheed.Sindhi last updated on 28/Oct/17

T han X a lot sir!