04 Apr 03
Following on from a problem set a while ago which I answered using Fermat's Little Theorem:
Find n composite such that (a,n) = 1 => a^n-1 = 1 (mod n) for all a.
(I don't know the answer, and my supervisor couldn't do it in the supervision at the time, saying I'm not an algebraist and looking something up in a book.)
05 Apr 03
Originally posted by AcolyteI thought about this all afternoon, and accomplished the following:
Following on from a problem set a while ago which I answered using Fermat's Little Theorem:
Find n composite such that (a,n) = 1 => a^n-1 = 1 (mod n) for all a.
(I don't know the answer, and my supervisor couldn't do it in the supervision at the time, saying I'm not an algebraist and looking something up in a book.)
1.I proved that such an n exists, and that there are more than one of them, but I would not wager that they are infinite in number;
2.I found a couple of examples of such numbers.
Let F(n):N->N be the number of integers less than or equal to n and relatively prime to it (the Euler totient function). A generalisation of FLT says that if gcd(a,n) = 1 then a^F(n) = 1 (mod n).
Then it can be shown that:
(a^f(n))^k = a^kF(n) = 1 (mod n)
For integers k. So all we need is some composite n such that F(n)|(n-1). I found necessary but not sufficient conditions that can be imposed on n for the above to be true, namely that n is an odd integer that is the product of at least three distinct primes, and is not divisible by the square of any prime. The proof of this is as follows:
To prove that it must be non-square-factorable, we realize that F is multiplicative and prove that n cannot be a square. So say n = p^2 for some prime p. Then F(n) = p^2 - p. So:
(n-1)/F(n) = [(p+1)(p-1)]/p(p-1) = (p+1)/p which is never an integer. So n may not have a aquare factor. Now if n has only two distinct prime factors, n = pq, then:
(m-1)/F(n) = (pq-1)/((p-1)(q-1) = 1 + 1/(q-1) + 1/(p-1) which cannot be an integer provided p and q are distinct. Thus some n satisfying your problem do exist, and are products of at least three distinct odd primes. I then fooled around with triples of small odd primes, and ruled out ones that failed to satisfy the condition F(n)|(n-1). The smallest one should be n = 3*11*17 = 561. But I am not an algebraist.
05 Apr 03
Originally posted by royalchickenCertain parts of this are wrong or not applicable. For example, on closer inspection, F(561) = 320 which does not divide 560. And it shouldn't, for I used a similar argument to prove that for F(n)|n-1, n needs at least FOUR prime factors. So the criteria is not that F(n)|n-1. Instead, we keep FLT in mind and look at n-1-F(n). From this we can arrive at the REAL criteria:
I thought about this all afternoon, and accomplished the following:
1.I proved that such an n exists, and that there are more than one of them, but I would not wager that they are infinite in number;
2.I found a couple of examples of such numbers.
Let F(n):N->N be the number of integers less than or equal to n and relatively prime to it (the Eu ...[text shortened]... ondition F(n)|(n-1). The smallest one should be n = 3*11*17 = 561. But I am not an algebraist.
1.For every p|n, (p-1)|(n-1)
2.n is not divisible by any squares
3.n is odd and all prime factors are distinct
So, using small primes, the first few are 561, 1105, 1729. If you'd care to continue this sequence, please do. Apologies for the partly faulty post last night.
PS If each of the following factors is prime, it is easy to show that (6m+1)(12m+1)(18m+1) satisfies your problem. Can you prove it (1729 is the first of these.)
07 Apr 03
Originally posted by bbarrThanks. This was a good problem; my first criterion was a bit too strong, so I weakened it and that seems to work. Thank you for showing me wolfram.com, too...I shall have to look at it more.
These are carmichael numbers:
http://mathworld.wolfram.com/CarmichaelNumber.html