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-   -   51+163k (https://www.mersenneforum.org/showthread.php?t=25210)

enzocreti 2020-02-10 16:03

51+163k
 
numbers of the form 51+163k






conjecture


51+163k is a square only for k=10

Batalov 2020-02-10 17:04

[CODE]for(k=1,10^10,if(issquare(51+163*k),print(k)))
10
91
255
498
826
1231
1723
2290
2946
3675
4495
5386
6370
7423
8571
9786
11098
12475
13951
15490
17130
18831
20635
22498
24466
26491
28623
30810
33106
...
...
[/CODE]

Yes infinitely many lol

NHoodMath 2020-02-10 23:39

Enzo, here is some relevant maths to what you have posted here:

The number 51, in your case, you were testing to see whether it was a quadratic residue (mod 163), that is, whether or not there is a square that is 51 (mod 163). There is a well-known theorem that states that if a number b is a quadratic residue (mod N), then there are infinitely many squares that are b (mod N). In this case, you found that for k=10, there is a square of the form k*N+b for N=163, b=51, therefore 51 is a quadratic residue (mod 163), and there are infinitely many squares of this form using the above theorem.

Batalov 2020-02-12 16:50

[QUOTE=Batalov;537236][

Yes infinitely many lol[/QUOTE]

[COLOR="DarkRed"]@ enzocreti - Do not edit other peoples posts.[/COLOR]


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