odd integer, squared , minus 1 ... the result is a multiple of _______ !

[henh...@gmail.com]

Take an odd integer, square it, and subtract 1 ( One )... then
the result is always a multiple of ____________ !




------- pls PLEASE wait a few days before posting answers or hints.





(Sum of 2 squares)
Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2


We've known about Squares ( and Pythagorean triples )
for so long that.... When we learn something
simple and new about them, it's like.... ___________

[Edward Murphy]

On 7/20/2022 8:33 AM, henh...@gmail.com wrote:

> Take an odd integer, square it, and subtract 1 ( One )... then
> the result is always a multiple of ____________ !

[spoiler space]































For any odd integer x:

x = 2y + 1 (for some integer y)

x^2 = 4y^2 + 4y + 1

x^2 - 1 = 4y^2 + 4y
= 4 * y * (y+1)

Obviously it's divisible by 4, but also either y or y+1 is divisible by
2 (if one is odd then the other is even), so x^2 - 1 is actually always
divisible by 8.

Sanity check:
1^1 - 1 = 0 = 8 * 0
3^1 - 1 = 8 = 8 * 1
5^1 - 1 = 24 = 8 * 3
7^1 - 1 = 48 = 8 * 6
9^1 - 1 = 80 = 8 * 10
and we can recognize the rightmost terms as
y * (y+1) / 2
for y = 0, 1, 2, 3, 4, etc.

[Richard Heathfield]

On 24/07/2022 9:41 pm, Edward Murphy wrote:
> On 7/20/2022 8:33 AM, henh...@gmail.com wrote:
>
>> Take an odd integer,  square it, and subtract   1 ( One )...
>> then
>>                 the result is always a multiple of
>> ____________  !
>
> [spoiler space]

Spoiler sans space: the answer is 1.

TOO easy.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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