Number theory

Prove that, the sequence 11,111,1111,11111,...âˆž doesn't have any perfect square.

5 Answers

341

Hari Shankar ·2009-07-07 08:58:16

the members of the sequence are all of the form 4k-1, and no perfect square is of the form 4k-1

Now how would i show that (4k-1) is never a perfect square?

Its a std result, squares are of the form 4k or 4k+1.

106

Asish Mahapatra ·2009-07-15 03:09:24

If u want its derivation,

A number is of the form 2n+1 or 2n

CASE 1: it is of form 2n
then x=2n
x² = 4n² = 4(k)

CASE 2: it is of the form 2n+1
then x=2n+1
x² = 4n²+4n+1
= 4(n²+n)+1
= 4(k)+1

So a square no. is of the form 4n or 4n+1

Thanks Asish ....

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