[ 3 / biz / cgl / ck / diy / fa / ic / jp / lit / sci / vr / vt ] [ index / top / reports ] [ become a patron ] [ status ]
2023-11: Warosu is now out of extended maintenance.

/sci/ - Science & Math

View post   
File: 460 KB, 1713x1012, day_50.png [View same] [iqdb] [saucenao] [google]
10579894 No.10579894 [Reply] [Original]

[math]
\text{Prove that there exist infinitely many integers }n\text{ such that }n,n+1,n+2\text{ are}
\\
\text{each the sum of the squares of two integers. [Example: }0=0^2+0^2\text{, }1=0^2+1^2\text{, }
\\
2=1^2+1^2\text{.]}
[/math]

>> No.10579895

Previous Thread >>10578703

>> No.10579935

>>10579894
n = 4 k^4 + 4 k^2
for any k.

>> No.10579942

Consider
n^2 - 1, n^2, n^2 + 1
the first factors as (n-1)(n+1) and is a sum of two squares if both (n-1) and (n+1) are sums of two squares.
So if (n-1), n, (n+1) is an example, then
n^2 - 1, n^2, n^2 + 1 is again an example and so we can generate infinitely many of them, provided we dont start with 0, 1, 2 since that will net 0, 1, 2 again, so start with like 8, 9, 10 or something.

>> No.10579947 [DELETED] 

>>10579894
[math]Prove~that~the~non-trivial~zeroes~of~the~Riemann~zeta~function~lie~on~Re(\frac12).[/math]

>> No.10580036

>>10579894
[math]n^2 + i^2[/math]
[math]n^2 + 0^2[/math]
[math]n^2 + 1^2[/math]

Get on my level faghots

>> No.10580208 [DELETED] 

>>10579935
>(n-1)(n+1) is a sum of two squares if both (n-1) and (n+1) are sums of two squares
why?

>> No.10580211

>>10579942
>(n-1)(n+1) is a sum of two squares if both (n-1) and (n+1) are sums of two squares
why?

>> No.10580435

>>10580211
it's well known, google it

>> No.10580688

bump

>> No.10580765

>>10580211
>>10580435
Because
(a^2+b^2)(c^2+d^2)
=(ac+bd)^2+(ad-bc)^2

>> No.10580774

>>10580765
oh, I see

>> No.10581464

>>10580765
That's pretty cool.

>> No.10581758

>>10580765
AC/DC?
Thunderstruck