/sci/ - Science & Math

>Be me
>598 years old alien computer scientist
>Testing a new automated theorem proving algorithm
>What we do is get a bunch of powerful computers, stitch them together, then start a new universe inside of them
>One century in the computer universe is a millisecond for us
>The goal is for the universe to eventually develop societies that will produce mathematicians and then start producing billions of generations of mathematicians, one of whom we hope will prove the Xylark Conjecture (what you would call the Riemann Hypothesis), or other important conjectures.
>I just got an error message that says one of the planets in my universe is wasting their time masturbating to anime girls and generally slacking off

Okay, listen up you pieces of shit. It wouldn't be the first time I put a black hole right next to the planet of a society that instead of working decided to fuck around. You are on the verge of getting destroyed. Right now you are wasting my computer time and my money. Every moment you spend masturbating is a moment my university is paying for electrical fees and maintenance fees so that I can keep my simulation. Did you hear that? Good. Now go back to being productive mathematicians or I am plugging off your entire sector of the universe. Why can't you be more like the Gorgorians from sector ZY521?

>>9140680
you must be 18+ to post on this board

>>9116773
BUT THE SECOND LAW OF THERMODYNAMICS SAYS THAT IF JEEESUUUUUUS DOESN'T EXIST THEN THE UNIVERSE DOESN'T EXIST.

THANKS JESUUUUUUUUUUUUUUUUUS

>mfw there are still christians in fucking 2017

>>8963411
>No mathematicians
Space was gay anyways, I don't need this shit.

>>8920246
>>8920224
>I'm a mathemusician

>>8885251
>you would be hailed as a hero in your hometown

This makes me feel really sad.
>tfw you would also get 100% probably
>tfw you already got 100% in every test you've taken since forever
>tfw because I don't live in the educated and responsible culture of China, no one sees me as a hero. Just as some retarded nerd.

First I want to make something clear. In this thread I will be discussing a problem but this is not a homework thread. I do not want nor am I asking for a solution. I want guidance. A discussion about how this problem may be approached and if there is someone knowledgeable enough to give references to books or articles that can help me out then that would be great too. I genuinely want to do this myself but I come here because after exhausting all of my knowledge I have come to the conclusion I am missing something fundamental needed to tackle this problem and I don't know where else to look for help. This is the problem:

[math]\text{Prove there is a constant } \epsilon > 0 \text{ with the following property:} \\ \text{If a,b,n are positive integers such that gcd(a+i,b+j) > 1 for every i, j} \in \{1,2,...,n\} \text{ then } \min \{ a,b \} > (\epsilon n)^n [/math]

Partial progress:
I have found and confirmed through various sources that for n=1, the smallest possible [math] \min \{ a, b\} [/math] is 14 and for n=2 it is 104.

They form the pairs (14,20) and (104,6200).
Other pairs I have found for the case n=2 are:
230 5654
494 5300
594 3128
644 5718
650 5704
664 4730
740 4654
740 6992
968 6764
1000 3794
1000 5564
1000 5654
1064 6460
1274 1308
1274 6408
1274 6698
1308 1274
1448 2714

My current understanding of the problem:
The problem is phrased in a way that makes you think [math] \epsilon [/math] have to be very small and that you probably have to advance through contradiction but what I have found with my examples is that [math] \epsilon [/math] does not need to be that small because [math] \min \{a,b \} [/math] grows really fast as you increase n.

For the case n=2 you can see that n^n will simply be 4 while the smallest [math] \min \{a,b \} [/math] is 104, way bigger.

That is why I think this has to be a direct proof that will involve approximating [math] \min \{a,b \} [/math].

Hopefully you can lend me a hand and this thread will be remembered as good.