Special whole numbers...

[cheesehead]

Factoris (http://wims.unice.fr/wims/wims.cgi) can factor not only integers (278914005382139703576000 = 2⁶ × 3⁵ × 5³ × 7² × 11 × 13 × 17 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 47), but also rational numbers, polynomials, and rational functions. It uses

"GP/PARI CALCULATOR Version 2.2.10 (alpha) for factorizations other than multivariate polynomials, and

Maxima version: 5.9.1 for factorizations of multivariate polynomials."

Factoris is part of the WWW Interactive Multipurpose Server (WIMS) at University of Nice. See http://wims.unice.fr/

[wpolly]

196883 is the minimal dimension of the faithful irreducible representations of "The Monster" , the largest sparodic finite simple group (its order is roughly 8E53).
Also, the so-called "Laurent series" of the j-function begins with:

j(t)=1/t+744+196884t+.......

Originally Posted by mfgoode

Wpolly: you are slightly off the mark. There is at least one and maybe more 13 digit prime(s) less than 1,000,000,000,063.
hint: you wont have to go far to find it!

Your number 196883: =196,560 + 323
196,560 is the number of spheres touching any one sphere in a 24-dimensional Leech lattice
323 = 17*19
However let's have your answer wpolly so the number is up again

196883.
Mally

The next number: 2478782

[wpolly]

Originally Posted by fetofs

EDIT: But, as we expect her answer (which probably is the one Kees gave):

I AM MALE..........

[mfgoode]

Originally Posted by cheesehead

Factoris (http://wims.unice.fr/wims/wims.cgi) can factor not only integers (278914005382139703576000 = 2⁶ × 3⁵ × 5³ × 7² × 11 × 13 × 17 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 47), but also rational numbers, polynomials, and rational functions. It uses

"GP/PARI CALCULATOR Version 2.2.10 (alpha) for factorizations other than multivariate polynomials, and

Maxima version: 5.9.1 for factorizations of multivariate polynomials."

Factoris is part of the WWW Interactive Multipurpose Server (WIMS) at University of Nice. See http://wims.unice.fr/

Thank you Cheesehead you are right on target. Thats the information I need:
'To build, not to destroy'
I have used factoris, the basic factorisation ,but I didnt explore any further
Mally

[mfgoode]

Originally Posted by wpolly

196883 is the minimal dimension of the faithful irreducible representations of "The Monster" , the largest sparodic finite simple group (its order is roughly 8E53).
Also, the so-called "Laurent series" of the j-function begins with:
The next number: 2478782

rank / prime / digits / who / when comment
1 / 2^ [(22478785)^(3)] +1 / 746190 / g245 / Oct 2003 / Divides

Fermat F(2478782), GF(2478782,3), GF(2478776,6), GF(2478782,12)
Mally

[fetofs]

Originally Posted by fetofs

EDIT: But, as we expect his answer (which probably is the one Kees gave):

278914005382139703576000

You were almost there for my number. This is 2^6×3^5×5^3×7^2×11×13×17×19×23×29×31×37×41×43×47, and is the smallest number with over a million distinct factors (1032192 to be exact).

[mfgoode]

Originally Posted by fetofs

You were almost there for my number. This is 2^6×3^5×5^3×7^2×11×13×17×19×23×29×31×37×41×43×47, and is the smallest number with over a million distinct factors (1032192 to be exact).

278914005382139703576000

:surprised A million distinct factors?
fetofs: you are a master indeed and someone to reckon with in dealing with numbers. .
Since this number can be represented by 15 different prime factors then the other factors must be combinations of these 15.
So starting with nC1 + nc2 +.... how far can one go to arrive at 1032192 to be exact ?
Mally

[garo]

The number of factors of any number that has a prime factor representation of:

p_1^n_1 * p_2^n_2......p_r^n_r

where p_1,p_2,....p_r are it's prime factors and n_1,n_2,...n_r are their respective exponents, is:

(n_1+1)*(n_2+1)*.....(n_r+1).

This includes 1 and the number itself, so you can remove those two if you do not wish to count them. So for
2^6×3^5×5^3×7^2×11×13×17×19×23×29×31×37×41×43×47
the total number of factors would be:

7*6*4*3*2^11 = 1,032,192

It is easy to see why the number of factors follows the relation given above. Note that each factor can have p_i in it from 0 to n_i times for a total of (n_i +1). See C&P ch. 1 for more details.

[mfgoode]

Thanks a million Garo. You explained it so brilliantly and lucidly that I had no problem to follow through.

I don't have C&P but Im sure my other number theory books will have it.
Your last para where the exponent is increased by 1 is a bit tricky but I'll fathom it out.

The notation you used is also self explanatory. After reading so many posts in
the forum and studying them I am able to not only understand it (the notation) but perhaps use them in my posts should the need arise.

I wish the other expert pollsters could be as simple.

I know there are posts explaining the notation used but I for one would welcome a comprehensive revival of such a topic for newbies like myself and others following this complex language in Number theory.
Thanks once again.
Mally

[garo]

Glad I could be of help. The notation I am using is the standard LateX math notation where _ is used for subscripts and ^ for superscripts. Let me give you an example to make things clearer.

30 = 2*3*5 so should have 2*2*2 = 8 factors. The factors of 30 can be formed by taking 0 or 1 occurence of 2,3 and 5 and hence there are 2*2*2 possible factors since there are two possibilities for each of 2,3 and 5. In the list below the first three numebrs refer to the number of occurences of 2,3,5 in the factor. So here we go:

0,0,0 -> 1
1,0,0 -> 2
0,1,0 -> 3
0,0,1 -> 5
1,1,0 -> 6
1,0,1 ->10
0,1,1 ->15
1,1,1 -> 30

As an exercise try this on 24. Not that 24 = 2^3 * 3^1 so we expect (3+1)*(1+1) = 8 factors.

[mfgoode]

Excellent Garo! Crystal clear and thank you.
You are as dedicated as your beloved father, my fellow contemporary.
God Bless you
Mally