 2:14 AM
I'm back everyone
But... I managed to find a few formulas which are quite interesting :)
$$(1\pm 3a)^3+(12+6a)^3=(9\pm 3a)^3+(10+6a)^3+(2a)^3$$ is one of them hehe
Notice that when $a=0$ we have $1729$, the smallest number which can be written as the sum of two positive cubes in two different ways :P
Anyway, Christmas is coming up.
How has everyone been lately? I've kinda ghosted this chatroom xD

2 hours later… 4:39 AM
Hello man!
@MrPie tis nice heh
btw question upvotes are now +10, hence the increase in your rep 5:18 AM
Wait what?
Sweet! :D

5 hours later… 10:38 AM
@OmegaKrypton welp looking back on it, I made a mistake when simplifying it :P Just change $3a$ to $9a^3$, $6a$ to $18a^3$ and $2a$ to $6a^3$ :)
No change $2a$ to $6a$ facepalm 11:29 AM
lel

8 hours later… 7:34 PM
@MrPie Oh wow, how did you find them? @TheSimpliFire I found a way to run at least pari/gp in ubuntu. I am not sure whether yafu and pfgw work under ubuntu as well.
Do you want to help search primes of the form (kn)! + n! + 1 with positive integers k,n ? Just send the routine for(k=1,100,for(n=1,100,s=(k*n)!+n!+1;if(ispseudoprime(s)==1,print([k,n])))) I get plenty OK, let us only consider k,n >= 100 ...
for(k=100,1000,for(n=100,1000,s=(k*n)!+n!+1;if(ispseudoprime(s)==1,print([k,n]))))
Maybe, we should first check that s is not too large ...
for(k=100,1000,for(n=100,1000,s=(k*n)!+n!+1;if(s<10^10^5,if(ispseudoprime(s)==1,print([k,n]))))) 7:51 PM
@Peter You can try running that. I'm offline in an hour
Current nontrivial results are
[2, 3]
[2, 8]
[2, 13]
[2, 19]
[6, 7]
[6, 8]
[7, 11]
[11, 54]
[17, 71] Can you run yafu on 10^100+9^9 and report the factor in factordb when you have finished ? Can't run yafu   8:08 PM
I wonder where the users Enzo Creti , didgogns and Robert Frost are !
Any output beyond 17/71 ?
@TheSimpliFire

1 hour later… 9:32 PM
@TheSimpliFire I have my secrets ;)
Nah, when I hop on a computer where I can perform some actual LaTeX, I'll show you
I found heaps of formulas like that, even short ones like $a^3-(a-1)(a-4)^2=(3a+4)^2$ if I did that correctly 