# a too-short sequence related to Latin squares

Richard Guy rkg at cpsc.ucalgary.ca
Wed Sep 8 21:29:06 CEST 2004

```Pari took 4 secs to
produce
96260050927125657231057045653340232713369826309730222706933915414681058441
? factor(%)

[1907 1]

[3413191 1]

[681082357 1]

[1089775321 1]

[29409077641 1]

[408476406083 1]

[1658631171224660115434923 1]

On Wed, 8 Sep 2004, Hugo Pfoertner wrote:

> Brendan McKay wrote:
>>
>> * Brendan McKay <bdm at cs.anu.edu.au> [040908 14:18]:
>>> 96260050927125657231057045653340232713369826309730222706933915414681058441
>>
>> Interesting factorization:
>>   (408476406083) (1658631171224660115434923) (681082357) (29409077641)
>>   (1089775321) (3413191) (1907)
>> Took Maple about a minute.
>>
>> B.
>
> The factorization is what one expects for a randomly chosen number C of
> this size. As a rule of thumb the expected number k of  prime factors is
> k=ln(ln(C)) +- sqrt(ln(ln(C))). For C~=9.62*10^73 we get k=5.1+-2.3. The
> largest prime factor is expected to be < C^0.6065 in 50% of all cases,
> the second largest PF < C^0.2117, so we expect 39 and 16 digits for the
> top 2 factors. (From Knuth TAOCP vol. 2, pp. 383-384).
>
> Maple seems to be rather slow doing factorizations. Dario Alpern's ECM
> http://www.alpertron.com.ar/ECM.HTM (one of the better sequence
> producing tools in my opinion) took 6 seconds for the complete
> factorization + sum of 3 squares on an 800 Mhz Athlon.
>
> Hugo
>

```