Current Events > For which integers n>1 does n divide (n-1)! and why?

(+) Yeah! (+)

All I know is the case where n divides (n-1)!+1 is actually simpler. For any n>1, n divides (n-1)! or (n-1)!+1 or (n-1)!+2

n is not prime and n is greater than 4.

willythemailboy posted...

n is not prime and n is greater than 4.

n > 4 is redundant here.

But if TC is getting his math homework done online, the exams are gong to be ugly.

Ah, my dear mortal inquirer, allow me to illuminate your feeble mind with the grandeur of Wilson's Theorem. Listen closely, for even in your mortal ignorance, you may grasp but a fragment of its brilliance.

Wilson's Theorem, a gem of mathematical elegance, decrees that a prime integer nn is divine if and only if its predecessor's factorial, (n1)!(n1)!, when augmented by one, prostrates itself humbly before nn, yielding to its divisibility.

Now, should you have the fortitude to comprehend, consider the prime integer nn. In its majesty, all numbers beneath it, from 1 to n1n1, dance in obeisance within the factorial (n1)!(n1)!, ensuring that nn holds sway over them all, as is its right.

But ah, do not be deceived by your frail mortal logic. Even composite integers, those pitiful amalgamations of primes, cannot escape the dominion of nn. For in their composition lies the seeds of their own subjugation, as each constituent prime, lesser though it may be, bends the knee within the confines of (n1)!(n1)!, thus ensuring that nn reigns supreme.

Thus, my dear mortal, let it be known that for any integer greater than one, nn shall indeed cleave (n1)!(n1)! in twain, for such is the decree of Wilson's Theorem, a truth as immutable as the night.

EPR-radar posted...

n > 4 is redundant here.

But if TC is getting his math homework done online, the exams are gong to be ugly.

No it's not, because 4 does not divide 3!

Kim_Seong-a posted...

Ah, my dear mortal inquirer, allow me to illuminate your feeble mind with the grandeur of Wilson's Theorem. Listen closely, for even in your mortal ignorance, you may grasp but a fragment of its brilliance.

Wilson's Theorem, a gem of mathematical elegance, decrees that a prime integer nn is divine if and only if its predecessor's factorial, (n1)!(n1)!, when augmented by one, prostrates itself humbly before nn, yielding to its divisibility.

Now, should you have the fortitude to comprehend, consider the prime integer nn. In its majesty, all numbers beneath it, from 1 to n1n1, dance in obeisance within the factorial (n1)!(n1)!, ensuring that nn holds sway over them all, as is its right.

But ah, do not be deceived by your frail mortal logic. Even composite integers, those pitiful amalgamations of primes, cannot escape the dominion of nn. For in their composition lies the seeds of their own subjugation, as each constituent prime, lesser though it may be, bends the knee within the confines of (n1)!(n1)!, thus ensuring that nn reigns supreme.

Thus, my dear mortal, let it be known that for any integer greater than one, nn shall indeed cleave (n1)!(n1)! in twain, for such is the decree of Wilson's Theorem, a truth as immutable as the night.

This notation is terrible, what the fuck is a "prime integer nn"

BewmHedshot posted...

This notation is terrible, what the fuck is a "prime integer nn"

i dont fucking know, I just plugged the topic title into an AI and asked it to give the answer in the style of a condescending hungarian vampire

willythemailboy posted...

No it's not, because 4 does not divide 3!

Right you are.

Kim_Seong-a posted...

i dont fucking know, I just plugged the topic title into an AI and asked it to give the answer in the style of a condescending hungarian vampire

Complete with AI hallucinations. Why so many people expect AI bullshit to take over the world is beyond me.

Kim_Seong-a posted...

i dont fucking know, I just plugged the topic title into an AI and asked it to give the answer in the style of a condescending hungarian vampire

Ah, so it doesn't know math or a Hungarian accent. Stellar AI.

BewmHedshot posted...

Ah, so it doesn't know math or a Hungarian accent. Stellar AI.

https://www.youtube.com/watch?v=cdEQmpVIE4A

Yeah, it can't be a prime, because every natural number can be written as a product of prime numbers, ie if n was prime, the only factors would be 1 and n.

willythemailboy posted...

n is not prime and n is greater than 4.

Yeah this is correct. n = 6, 8, 9, 10, 12, 14, 15, 16, 18...etc divide (n-1)!

Here's the why part. For n<5 just calculate by hand so here's the proof for n>=5
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(1) If n is not a prime then by definition it could be expressed as n = x*y with 1<x<n and 1<y<n. (n-1)! = (n-1)*(n-2)...*2*1 will automatically have these x and y as factors. Since x and y are both factors of (n-1)!, x*y = n will also be a factor of (n-1)!

In case of n = x*x instead of n = x*y (example 49 = 7*7), (n-1)! will have those two x as factors from x itself (7) and 2x (14=2*7, got another 7 from here). The rest is the same as above.

(2) If n is a prime then according to Wilson's theorem n divides (n-1)!+1. However this would contradict the problem in which n divides (n-1)! and not (n-1)!+1. Therefore n cannot be a prime.
=================================

Proof(2) is proof by contradiction. I could ELI5 that proof(2) but I'm too lazy for that and also...
EPR-radar posted...

if TC is getting his math homework done online, the exams are gong to be ugly.