It began with a question: Are any Factorials an Exact Power? (The factorial of n is the product of all natural numbers less than or equal to n. An exact power is an n = m^s for natural numbers m,s>1.) Well? Guess first before we do any thinking. But guess No. Because anything else should make the maths part of your brain sad and hurt. So no. But also no, there won't be any thinking here. In that way lies unhealth and it's far too hard anyways. Rather, the modern tenor must be embraced. 'The Computer' shall do our thinking for us. This below runs out prime factor decompositions of some early factorials. [Sigh. I bet that's incorrectly sized. What a drag. But look, I tried, okay.] An exact power isn't too hard to spot from a prime decomposition. All the powers should be divisible by some common number greater than 1. A really easy case if is all the powers are the same, for instance. But you may have a case where all the powers are 2 say, except for the power on one prime which might be 4 perhaps. That's still an exact power because the prime, p, which is raised to the fourth power is still a square, only of itself squared, p^2. Because of (p^2)^2 = p^4. So we're looking to find a prime decomposition powers sequence with a nontrivial(I'd say decent) highest common factor. Now so, what does 'The Computer' say when this is run... First it tells us what primes it is worrying with. The engineer in all of us now declaims that the power of the highest prime in the decomposition of any factorial is 1. So there can't be an exact power n! = m^s with m,s>1. Because we must group up all the prime factors of n! to give the m if it exists. However the top prime won't ever be a square or higher power. So we won't be getting our s>1. Tough sh*t, really. All very helpful, thank you Mr. Computer. It's useful having you around when we don't have a brain of our own. Still, it nags. How should we have seen this originally if we were proper clever? The answer, of course, is Bertrand's Postulate. A fantastic piece of magic which I shall salute with my first pint of the evening later. In fact, you should always be thinking Bertrand's Postulate because it is the quick jackknife punch in the stomach for getting out of dark elementary number theory alleys. Now it says there's always a prime between n and 2n for n>1. So suppose the top prime, p, of our factorial is a square. (Or any higher power because cubes have squares in them, etc.) Then what happened to the prime, q, the one between p and 2p. It is larger than p but smaller than 2p. And if p^2 is in the prime decomposition of n! then n>=2p for how else is it to sweep up the doubled prime except with the p and 2p parts of the factorial multiplications at best. This leaves us with p the top prime in n! and n >= 2p > q > p. Meaning q is one of the terms in n! only it is not top prime despite being prime and larger that the top prime p. Which is all bogus. It's great not having to think, isn't it? Only is it? The great fear is of compthink groupthink. Has the computer tended us to a particular solution? One which doesn't speak to the real essence of the problem? Computer lead investigations have a natural computational flavour. Because that's what they are, computational. But a computational proof is an unseemly thing at times. They can be arithmeticisms and not mathematicisms. Mathematics wants 'real solutions' to questions, the solutions which enlighten and inform. And only occassionally are these computational style. But now the pedagogy of this example goes missing. Hey, don't hassle me, I didn't cook it up to straw man a moral position. The highest prime is critical to deciding that factorials are never exact powers. You are basically stuck with finding it a pairing factor to sweeten out into a power. And you'll only get that factor from higher up the factorial product. But we know that by the time you get to that factor, the stuff in between will have messed in a way that puts you back where you started. Namelu, looking for a complement to a new highest prime. For sure? Isn't there maybe some other explanation at play? Well, look at the decomposition of 10!. It's: Extra Credit: The decomposition profiles seem to grow strings of ones on the top side. Is this for sure? Will the number of single power primes at the top of a factorial prime decomposition grow arbitrarily large or is it gonna happen that they will shrink? And if they collapse to a single one, can all the factors but the top prime sort themselves into a convenient square or higher power arbitrarily often?
/**
* Class to examine the powers of factorials.
*/
import java.util.Arrays;
/**
* Print prime decomposition profiles for early factorials
*/
public class FactorialPowers
{
/**
* Short primes list for testing. Primes < 100
*/
private static final int[] PRIMES = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 };
/**
* @param args unused
*/
public static void main(String[] args)
{ // We build the factors of n! from (n-1)! and n. So we want to be
// keeping a tally of the primes seen each time through our loop.
// factors[i] is the power of PRIMES[i] in the prime decomposition
// of n! (Loop Invarient across all the indices)
int[] factors = new int[PRIMES.length];
// We start looping with the prime decomposition of 1, no prime factors.
// for (int i = 0; i < factor.length; i++) { factors[i] = 0; }
// For each factorial we are sure we can properly factor...
// i.e. as far as our last prime. Can go last prime +1 but not +2 since
// that itself can be prime and not already in our list.
// <= because we can go as far as the last prime okay.
System.out.println("Primes" + Arrays.toString(PRIMES));
for (int n = 2; n <= PRIMES[PRIMES.length - 1]; n++)
{ // We have already factorised (n-1)! so just add the factors of n.
// Run though our primes less than n pulling out factors,
// breaking down n as we go until it is 1.
int i = 0;
int n_ = n;
while (n_ > 1)
{ // Is the i-th prime a factor of n_? If so add it to factors,
// break down n_ and reenter the loop to test the i-th prime
// for a repeated factor
if (n_ % PRIMES[i] == 0)
{ factors[i]++;
n_ /= PRIMES[i];
}
else // Otherwise step on to testing the next prime.
{ i++;
}
// Some sanity...
// CAN'T HAPPEN: Our i-th PRIME exceeds n_ and we're not already done
assert (!(PRIMES[i] > n_ && n_ != 1));
// TEST: The i index should be in bounds or we're already done.
assert (i < PRIMES.length || n_ == 1);
}
// Dump the computed factors to the console.
System.out.println(n + "! : " + Arrays.toString(factors));
}
}
}
Primes[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
And then it tells us a bunch of happy decompositions.
2! : [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Because 2! = 2. Nicely.
3! : [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Because 3! = 2 * 3. Isn't this computer clever?
4! : [3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Because 4! = 2 * 3 * 4 = 2 * 3 * (2 * 2) = 2^3 * 3. And blah and blah for...
5! : [3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
6! : [4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
7! : [4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
8! : [7, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
9! : [7, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
10! : [8, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
11! : [8, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
12! : [10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
13! : [10, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
14! : [11, 5, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
15! : [11, 6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
16! : [15, 6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
17! : [15, 6, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
18! : [16, 8, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
19! : [16, 8, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
20! : [18, 8, 4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
21! : [18, 9, 4, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
22! : [19, 9, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
23! : [19, 9, 4, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
24! : [22, 10, 4, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
25! : [22, 10, 6, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
26! : [23, 10, 6, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
27! : [23, 13, 6, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
28! : [25, 13, 6, 4, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
29! : [25, 13, 6, 4, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
30! : [26, 14, 7, 4, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
31! : [26, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
32! : [31, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
33! : [31, 15, 7, 4, 3, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
34! : [32, 15, 7, 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
35! : [32, 15, 8, 5, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
36! : [34, 17, 8, 5, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
37! : [34, 17, 8, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
38! : [35, 17, 8, 5, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
39! : [35, 18, 8, 5, 3, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
40! : [38, 18, 9, 5, 3, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
41! : [38, 18, 9, 5, 3, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
42! : [39, 19, 9, 6, 3, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
43! : [39, 19, 9, 6, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
44! : [41, 19, 9, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
45! : [41, 21, 10, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
46! : [42, 21, 10, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
47! : [42, 21, 10, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
48! : [46, 22, 10, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
49! : [46, 22, 10, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
50! : [47, 22, 12, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
51! : [47, 23, 12, 8, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
52! : [49, 23, 12, 8, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
53! : [49, 23, 12, 8, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
54! : [50, 26, 12, 8, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
55! : [50, 26, 13, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
56! : [53, 26, 13, 9, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
57! : [53, 27, 13, 9, 5, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
58! : [54, 27, 13, 9, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
59! : [54, 27, 13, 9, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0]
60! : [56, 28, 14, 9, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0]
61! : [56, 28, 14, 9, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
62! : [57, 28, 14, 9, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
63! : [57, 30, 14, 10, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
64! : [63, 30, 14, 10, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
65! : [63, 30, 15, 10, 5, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
66! : [64, 31, 15, 10, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
67! : [64, 31, 15, 10, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
68! : [66, 31, 15, 10, 6, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
69! : [66, 32, 15, 10, 6, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
70! : [67, 32, 16, 11, 6, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
71! : [67, 32, 16, 11, 6, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
72! : [70, 34, 16, 11, 6, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
73! : [70, 34, 16, 11, 6, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
74! : [71, 34, 16, 11, 6, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
75! : [71, 35, 18, 11, 6, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
76! : [73, 35, 18, 11, 6, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
77! : [73, 35, 18, 12, 7, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
78! : [74, 36, 18, 12, 7, 6, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]
79! : [74, 36, 18, 12, 7, 6, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0]
80! : [78, 36, 19, 12, 7, 6, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0]
81! : [78, 40, 19, 12, 7, 6, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0]
82! : [79, 40, 19, 12, 7, 6, 4, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0]
83! : [79, 40, 19, 12, 7, 6, 4, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
84! : [81, 41, 19, 13, 7, 6, 4, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
85! : [81, 41, 20, 13, 7, 6, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
86! : [82, 41, 20, 13, 7, 6, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
87! : [82, 42, 20, 13, 7, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
88! : [85, 42, 20, 13, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
89! : [85, 42, 20, 13, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
90! : [86, 44, 21, 13, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
91! : [86, 44, 21, 14, 8, 7, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
92! : [88, 44, 21, 14, 8, 7, 5, 4, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
93! : [88, 45, 21, 14, 8, 7, 5, 4, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
94! : [89, 45, 21, 14, 8, 7, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
95! : [89, 45, 22, 14, 8, 7, 5, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
96! : [94, 46, 22, 14, 8, 7, 5, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]
97! : [94, 46, 22, 14, 8, 7, 5, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
10!=2.3.4.5.6.7.8.9.10=2.3.(2^2).5.(2.3).7.(2^3).(3^2).(2.5)
And we rearrange to get 10! = (2^8).(3^4).(5^2).7 and here our only problem is the highest prime, 7. The rest is:
(2^8).(3^4).(5^2)=(2^4.3^2.5)^2=720^2
So yeah, this is a case where the bottom stuff all sorts itself out and only the top prime causes grief.
18 January 2007
Bertrand's Postulate
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4 comments:
You are a very scary person sometimes.
You will have to explain this to me over a pint sometime cos you lost me at "Because we must group up all the prime factors of n! to give the m if it exists." It sounds right but frankly I cant quite see that its true.
I'm happy to discuss anything over a pint.
Quickly: But well, you're right really. I mean to say sometime more subtle and the wavy notion things in my head doesn't match exactly what is on the page. It isn't enough to just take the individual primes of the decomposition and stick them together to make your m. What I want to say is that you take them and possibly some powers of them.
What should be said is that if you come up with an m^s=n! for an s bigger than 1, then by considering the prime decomposition of n! you get into a problem. On the left it says the prime decomposition of n! is the prime decomposition of m with all the powers multiplied by s. And since s is greater than one, all those powers must be greater than 1.
However, we see later that at least one power on a prime in the prime decomposition of n! is always 1. Namely, the power of the largest prime in the decomposition.
What I wanted to mean with the group up fluffy sentence is that the m is formed out of the primes in the decomposition of n!. (Love the punctuation there.) I might have made clear that these might not be single powers. Like in the case of 10! ignoring the 7, we would have:
10! = 7 x (2^4 x 3^2 x 5)^2
Pretending the 7 isn't here, the m would be 2^4 x 3^2 x 5, not 2x3x5 as it would seem from the sentence.
I hate mathematics sometimes. You've something in your head and it's too subtle for what you put on the page in your hurry.
The pint, though, for sure.
I think I'm seeing it now - not in the sense that I could actually replicate it or anything but I'm with you as they say.
Loving the "Never Work" motto, BTW. Pithy.
It's the Parisians:
http://www.bopsecrets.org/CF/graffiti.htm
Thinking of making a pilgrimage there this year. If I'm lucky it'll be during the Presidential election and in the middle of a huge riot. One can hope...
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