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.