/sci/ - Science & Math

i'll bump with cats and after a while i'll give a hint

14^-1 doesn't exist for m=26, you can't find any number a' such that 14*a'=1 mod 26

ITT: math challenges. Post basic math questions that require witty or beautiful proof, or answer to someone else's

I'll start: prove there are infinitely many pairs of natural numbers (m,n) such that the sum of integers from 1 to m equals the sum for m to n.

>>6753875
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>>6751019

i'll give da proof anyway:
assume a and n being as you say: <span class="math"> \sum_{k=1}^a k = \sum_{k=a}^n k [/spoiler].
That is to say <span class="math"> \sum_{k=1}^a k = \sum_{k=1}^n k - \sum_{k=1}^{a-1} k [/spoiler].
It means <span class="math"> \frac{a(a+1)}{2} = \frac{n(n+1)}{2}-\frac{(a-1)a}{2} [/spoiler],
which also means that <span class="math"> \frac{n(n+1)}{2} = a^2 [/spoiler].

We are hence looking for an unbounded set of n such that <span class="math"> \frac{n(n+1)}{2} [/spoiler] is a squared natural number.
Let's build a sequence <span class="math"> n_k [/spoiler] of such numbers

let <span class="math"> a_0=1 [/spoiler] and <span class="math"> b_0=1 [/spoiler], and <span class="math"> a_{k+1}=a_k+b_k [/spoiler], <span class="math"> b_{k+1}=2a_k+b_k [/spoiler]
I say that (1):there is a <span class="math"> n_k [/spoiler] such that <span class="math"> (a_kb_k)^2=\frac{n_k(n_k+1)}{2} [/spoiler]
(then <span class="math"> n_k [/spoiler] works)

To prove (1), let's have an induction about k: there is a <span class="math"> n_k [/spoiler] such that either
(2):<span class="math"> a_k^2=\frac{n_k}{2} [/spoiler] and <span class="math"> b_k^2=n_k+1 [/spoiler], either
(3):<span class="math"> a_k^2=\frac{n_k+1}{2} [/spoiler] and <span class="math"> b_k^2=n_k [/spoiler]

First, (2)=>(1) and (3)=>1 is kinda obvious
Then, <span class="math"> a_0=1 [/spoiler] and <span class="math"> b_0=1 [/spoiler] satisfy (3) with <span class="math"> n_0=1 [/spoiler]

And the induction:
First case, if <span class="math"> a_k [/spoiler] and <span class="math"> b_k [/spoiler] satisfy (3) with <span class="math"> n_k [/spoiler],
then we have
<span class="math"> b_{k+1}^2=(2a_k+b_k)^2=4a_k^2+b_k^2+4a_kb_k=4\frac{n_k+1}{2}+n_k+4a_kb_k [/spoiler]
<span class="math"> b_{k+1}^2=2n_k+2+n_k+4a_kb_k=n_{k+1}+1 [/spoiler] if we call <span class="math"> n_{k+1}=3n_k+1+4a_kb_k [/spoiler]
<span class="math"> a_{k+1}^2=(a_k+b_k)^2=a_k^2+b_k^2+2a_kb_k=\frac{n_k+1}{2}+n_k+2a_kb_k=\frac{3n_k+1+4a_kb_k}{2}=\frac{n_{k+1}}{2} [/spoiler]
and we have property (2) for k+1 (...)

>>5443950
Ah, that explains your answer then. I should have been more explicit I suppose.

Hello, it's my first time posting on /sci/.
I was wondering, for no particular reason, is it possible to make an electromagnet out of water?

>>4727351
so the differing number in chromosomes also factors in those that were combined...

does this have anything to do with the 'reptilian, mammal, human' levels of the brain?

>>4637852
so then, what do you recommend, instead?