Strange factorization

[CRGreathouse]

I asked a question on stats.stackexchange about the factorization of 2015⁴ + 4¹³⁴⁵ (a number I just 'happened upon') because I was struck by the somewhat unusual factorization. At the time I was hoping for an algebraic factorization that I had missed, though this seems unlikely since 2015⁴ + x¹³⁴⁵ is irreducible. But is there any reason for this behavior? If it was just a typical number of its size the chance that it would have so many factors so (relatively) close together is something like .3% (which, I was reminded, corresponds to an alpha of about .006 since a priori I could have been surprised in either direction).

I did not cherry pick this number -- it was the only number I examined, and I suspected something funny -- algebraic factorization or other -- before I attempted the factorization.

It could be simple chance but I think not -- I think it shows a lack of understanding of factorizations on my part. Educate me!

[Batalov]

2015⁴ + 4*x⁴ is reducible, though...

[axn]

Yes. Looks like Aurifeuillean Factorization is at play. The 405-digit unfactored part and it's cofactor are very close together in size.

That still leaves the question of why one of the cofactors split further into so many.

[CRGreathouse]

Originally Posted by Batalov

2015⁴ + 4*x⁴ is reducible, though...

Perfect! That's why I love this forum.

Originally Posted by axn

Yes. Looks like Aurifeuillean Factorization is at play. The 405-digit unfactored part and it's cofactor are very close together in size.

That still leaves the question of why one of the cofactors split further into so many.

Indeed.

[R.D. Silverman]

Originally Posted by CRGreathouse

Perfect! That's why I love this forum.



Indeed.

Nomenclature correction:

It is not an Aurefeuillian factorization. i.e. that of X^4 + 4Y^4

Apply Erdos-Kac. How many factors does each of the algebraic
factors have? Is it more than 3 Sigma from the mean?

[Batalov]

When I first looked at the factordb entry it still had a c650 cofactor. But I recently was ruminating about x^y+y^x and convinced myself that x=4 would be a "Sierpinski-like number" for it because the expression was never prime (y>1), algebraically. Well, it is not a "Sierpinski-like number" in spirit, really; there is no covering set.

So, I submitted the
2015^2+2*4^672+2*2015*2^672
2015^2+2*4^672-2*2015*2^672
factors; the DB usually does gcd, but it didn't. Then I ran gcd in Pari and submitted the c245 and c405, and the entry started to look like it does now.

For fun, I've done the same to
2015^4+4^1015
2015^4+4^2015
Of course, one can also generate a test file of these algebraic factorizations with awk or perl and submit it to the DB...

[CRGreathouse]

Originally Posted by R.D. Silverman

Apply Erdos-Kac. How many factors does each of the algebraic
factors have? Is it more than 3 Sigma from the mean?

2^672*4030+4^672*2+2015^2

I don't have a full factorization, so all I can say is that it has 8 or more prime factors. 8 wouldn't be unusual for a number of that size. The other algebraic factor is completely unfactored.

[Batalov]

9 factors, after all.

[CRGreathouse]

Originally Posted by Batalov

9 factors, after all.

So that's definitely unusual clustering on the one algebraic factor. Does anyone know why? I see that 44971818273701332261784061961 * 9664021418404865297256058765601 * 386265978137298005895635792872544753829637 is close to a quarter of the logarithmic total, but not close enough that I could reasonably expect something nice like the original factorization.