Math Is Fun Forum

Emo

Guest

prove that !!

Hii !!

prove, for all n positive integer, That :

= The Floor of x .

Can you help me !

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: prove that !!

Note that neither

nor

is ever a perfect square as all even perfect squares must be divisible by 4 and all odd perfect squares must be

. In other words there are no perfect squares

such that

, i.e. there are no integers

such that

.

It follows that

.

To complete the proof, show that

Last edited by JaneFairfax (2010-01-30 05:45:35)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: prove that !!

Nice!!
On the same line, for beginners,
notice that the series of squares: {1,4,9,16,25,36,49}
can be obtained by starting with 0, and doing the following
thing with 4's.  Got this idea from Jane above:
Add increasing odd digits to obtain the next square.
0 --> 1 by adding 0*4 + 1  (this +1 is also 0 + 1)
1 --> 4 by adding 1*4 - 1  (this +3 is also 1 + 2)
4 --> 9 by adding 1*4 + 1 (this +5 is also 2 + 3)
9 --> 16 by adding 2*4 - 1 (this +7 is also 3 + 4)
16 --> 25 by adding 2*4 + 1 (this +9 is also 4 + 5)
25 --> 36 by adding 3*4 - 1 (this +11 is also 5 + 6)
36 --> 49 by adding 3*4 + 1 (this +13 is also 6 + 7)
(hope this post doesn't detract from the more important one above)

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