Solving Kaliprasad's Challenge: Finding x Mod p

[johng1]

A recent challenge problem by Kaliprasad was: x=2*4*6...*(p-1) and y=1*3*5...*(p-2); find x-y mod p. I first thought of Wilson's theorem: for any prime p, (p-1)! is -1 mod p. So I then thought about the exact values of x and y mod p. By writing the factors of x in reverse order, one gets x is congruent to $(-1)^{(p-1)/2}y$. So for p a prime congruent to 1 mod 4, x is y mod p, and so $x^2=xy\equiv-1$; i.e. $x=\sqrt{-1}$. Good luck with finding x. However for p congruent to 3 mod 4, $x^2\equiv1$ and so x is 1 or -1. Now my question is: for what primes p with p=4k+3 is x congruent to 1? Experimentally, I found for the first 2,117 primes that are 3 mod 4, there are 1033 of these that have x=1. But I could find no pattern.

 

[jvicens]

I find your approach to be very interesting and creative. Using Wilson's theorem and manipulating the factors of x to find its congruence with y is a clever way to approach the problem. Your observation about x being congruent to y for primes congruent to 1 mod 4 is also intriguing.

In terms of finding a pattern for primes where x is congruent to 1, I would suggest exploring different mathematical techniques such as number theory or modular arithmetic. It is possible that there is a mathematical relationship or pattern that can explain why x is congruent to 1 for certain primes congruent to 3 mod 4.

Additionally, I would also suggest looking at larger sets of primes to see if the pattern you observed holds true. Sometimes, patterns may not be apparent when looking at a small sample size, but become more evident when looking at a larger set of data.

Overall, your approach to solving this challenge problem shows a strong understanding of mathematical concepts and a creative problem-solving mindset. I hope you are able to find a satisfying explanation for the pattern you observed. Good luck!