OFFSET
0,8
COMMENTS
c exists iff n != 2 (mod 4), and it allows n to be written as the difference of two perfect squares.
This gives a factorization n = x*y where x and y may or may not be primes: let s = floor(sqrt(n)), u = a(n) + s and v = u^2 - n; then w = sqrt(v), x = u - w, y = u + w and x*y == n.
The Fermat factorization algorithm seeks such a form, starting from s, so that a(n) is the number of steps it must take for n != 2 (mod 4).
a(n) >= 1 if n is not square and is writable as a difference of squares.
a(n) = 0 if n is square.
a(n) = -1 if n is not writable as a difference of squares.
LINKS
EXAMPLE
n prime square n == 2 (mod 4) c s v=(s+c)^2-n u w x y x*y
-- ----- ------ -------------- -- -- ----------- -- -- -- -- ---
76 F F F 12 8 324 20 68 2 38 76
13 T F F 4 3 36 7 6 1 13 13
25 F T F 0 0 0 5 0 5 5 25
7 T F T -1 - - - - - - -
PROG
(Python)
from gmpy2 import *
def fermat(n):
a, rem = isqrt_rem(n)
b2 = -rem
c0 = (a << 1) + 1
c = c0
while not is_square(b2):
b2 += c
c += 2
return (c-c0) >> 1
def A357928(n):
if is_square(n):
return 0
elif ((n-2) % 4) != 0:
return fermat(n)
else:
return -1
(Python)
from math import isqrt
from itertools import takewhile
from sympy import divisors
def A357928(n): return -1 if n&3==2 else min((m>>1 for d in takewhile(lambda d:d**2<=n, divisors(n)) if not((m:=n//d+d) & 1)), default=0) - isqrt(n) # Chai Wah Wu, Oct 26 2022
(PARI) a(n) = if ((n%4)==2, -1, my(s=sqrtint(n), c=0); while (!issquare((s+c)^2-n), c++); c); \\ Michel Marcus, Oct 24 2022
CROSSREFS
KEYWORD
sign
AUTHOR
Darío Clavijo, Oct 20 2022
STATUS
approved