"Le raisonnement par récurrence [...] est un instrument qui permet de passer du fini à l’infini [...]" – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

"Le raisonnement par récurrence [...] est un instrument qui permet de passer du fini à l’infini [...]" – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

Breast cancer recurrence
Hormone receptor status clues
Stage may foretell fate

#breastcancer #recurrence #hormonereceptor #haiku #poetry

http://www.medicalnewstoday.com/articles/breast-cancer-recurrence-could-cancer-stage-and-receptor-status-help-predict-risk

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence

So, the first four pseudoprimes generated by the #recurrence sequence
a(n) = a(n-1) - a(n-2) - a(n-4), starting with (4, 1, -1, -2),
(that is, composite numbers passing the probable-prime test a(p) ≡ 1 mod p)
are, unless I have slipped up somewhere, 2706, 5205530, 42161779, 1146180898. That's a big gap before number four!

Here's a new one of these prime-detecting #recurrence sequences for you:

a(n) = a(n-1) - a(n-2) - a(n-4)
Start with 4, 1, -1, -2.
For prime p, a(p) ≡ 1 (mod p).

My search in 2013 passed over this because the first pseudoprime is just 2706, but the second one is 5205530 and the third is > 40 million.

The other good fourth-order ones with coefficients in {+-1, 0} all seem to be basically the Perrin case again (characteristic polynomial is a multiple of the Perrin one).

@johncarlosbaez @ColinTheMathmo These methods could definitely be applied to the whole family of sequences I've been messing with. The fastest method I knew of was Joerg Arndt's approach of calculating large powers of matrices over Z_n, since we only need the moduli--PARI/GP is way better at this than sagemath--this is pretty good but does get harder as the numbers get truly immense. #recurrence

But the analysis having to do with PV numbers doesn't apply to Lehmer's number, since it is a "Salem number": most of the other roots are ON the unit circle and will keep wreaking havoc with the moduli arbitrarily far out. So I understand that less. It is conjectured to be the smallest Salem number. #recurrence

Anyway, what got me started on all this again was that a post here that I can no longer find mentioned "Lehmer's number", the leading root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1. Naturally I wondered if the recurrence corresponding to this polynomial was a good prime detector. It is: the only pseudoprime less than, I think, 4 million is 2014. But it turns out that 10th-order recursions that good are not all that hard to find, so it isn't super special in that respect. #recurrence

So I guess all of this gives us a generalization of the Pisot-Vijayaraghavan numbers beyond the positive reals, and I'm sure someone has studied this already, but I haven't seen it. #recurrence

I also found a lot where the leading roots are a conjugate pair of complex numbers, and there, if you add their powers, you get this sort of exponentially exploding sinusoidal oscillation--but (if the other roots have norm < 1) one where all the values get exponentially close to integers! That's real weird. #recurrence

It is, by the way, profoundly strange to see this exact expression where we raise a bunch of generally irrational and/or complex numbers to large powers, add them together, and it turns out to just be *rounding* the leading term to an integer.

But that's a remarkable property of these PV numbers--they get close to integers when you raise them to large powers. But the PV numbers are just the positive real ones--for my fifth-order recurrence, the leading root is negative. #recurrence

Anyway, if you look at that Binet-like formula, you can see that the sum should pretty quickly be dominated by the root(s) of largest norm.

But we're doing modulo arithmetic on the sums, so the smaller terms will still be important--but *if* they all have norm less than 1, pretty soon all they're doing is rounding the sum to the nearest integer. But early on, they can be doing more than that. I think that is what causes some of the small-number weirdness. #recurrence

But as yet, I have no good feel for what makes a "good" sequence whose probable-prime test passes few composites, preferably with a few small-integer coefficients in the recurrence.

The polynomials associated with the Lucas and Perrin numbers are famous: their leading eigenvalues are the golden ratio and the "plastic number" respectively, small Pisot-Vijayaraghavan numbers as I was saying. But this doesn't lead to a simple pattern for "good" sequences. #recurrence

Some other relevant facts: Sequences of this type all have a simple closed-form "Binet-like" exact expression for their elements:

\[ a_n = \sum_{j} c_j^n \]

where the c_j are the eigenvalues of the matrix M. Those are also the roots of a monic polynomial that comes from det (Ix - M) = 0. Basically the leading coefficient is always 1 and the others are minus the recurrence coefficients. So these recurrences seem to have a lot to do with the study of algebraic integers. #recurrence

You don’t understand “You won’t understand until you’re older” until you’re older.

#theshowerthoughts #recurrence