I do this thing sometimes where I’ll just make stuff up, stick it in scipy and see what happens. I rarely share it publicly, because my ego is intensely fragile and allergic to public exposure. Onward…

import numpy as np

def pmatrix_rptcol():
‘’’
create matrix where first row is primes,
first column is repeated 2
‘’’
pvec0 = [i for i in range(30) if isprime(i)]
def make_prime_row(pvecn):
ret_pvec = list(range(len(pvecn)))
for i in range(len(pvecn)):
if i == 0:
ret_pvec[i] = pvecn[i]
else:
ret_pvec[i] = pvecn[i] - pvecn[i-1]
return ret_pvec
pvec1 = make_prime_row(pvec0)
pvec2 = make_prime_row(pvec1)
pvec3 = make_prime_row(pvec2)
pvec4 = make_prime_row(pvec3)
pvec5 = make_prime_row(pvec4)
pvec6 = make_prime_row(pvec5)
pvec7 = make_prime_row(pvec6)
pvec8 = make_prime_row(pvec7)
pvec9 = make_prime_row(pvec8)
return np.array([pvec0, pvec1, pvec2, pvec3, pvec4, pvec5, pvec6, pvec7, pvec8, pvec9])

All this means is, there is a matrix, where a number is equal to:

the number above minus the number to the left of the number above it

except for column 1. they are all 2. i don’t know why. why not?

It’s a 10x10 invertible matrix of integers, with independent columns where P != P.T (i.e. not symmetric, or not equal to its transpose, if it’s been a while)

I found this handy function to tell if a number is prime or not. I literally could not be bothered writing one myself at this time of night, so stack overflow helped me:

def isprime(n):
‘’’
check if integer n is a prime
shamelessly ripped from:
http://stackoverflow.com/questions/18833759/python-prime-number-checker#18833845
‘’’
# make sure n is a positive integer
n = abs(int(n))
# 0 and 1 are not primes
if n < 2:
return False
# 2 is the only even prime number
if n == 2:
return True
# all other even numbers are not primes
if not n & 1:
return False
# range starts with 3 and only needs to go up
# the square root of n for all odd numbers
for x in range(3, int(n**0.5) + 1, 2):
if n % x == 0:
return False
return True

You just do this:

In [86]: isprime(12430987123409872134098721340987530987321409872130948721094872134098721340987213409872134098721398765984310968431609814369867)
Out[86]: False

It even works with big numbers!

There is also this matrix, because I thought it a touch silly to have a column of just 2’s everywhere…

def pmatrix_norptcol():
‘’’
create matrix of first row prime,
first column not repeated
‘’’
pvec0 = [i for i in range(30) if isprime(i)]
pvec1 = [i-1 for i in pvec0]
pvec2 = [i-1 for i in pvec1]
pvec3 = [i-1 for i in pvec2]
pvec4 = [i-1 for i in pvec3]
pvec5 = [i-1 for i in pvec4]
pvec6 = [i-1 for i in pvec5]
pvec7 = [i-1 for i in pvec6]
pvec8 = [i-1 for i in pvec7]
pvec9 = [i-1 for i in pvec8]
return np.array([pvec0, pvec1, pvec2, pvec3, pvec4, pvec5, pvec6, pvec7, pvec8, pvec9])

And then we try all the usual decompositions, basis, eigenvalues, etc etc…

I’ll post back with more once I’ve got my data in a presentable state.