special_factorization_identity.sf

#!/usr/bin/ruby

# Daniel "Trizen" Suteu
# Date: 09 August 2019
# https://github.com/trizen

# A new special factorization identity for:
#
#    a(n) = (2^f(n) - f(n) + 1) * 2^f(n) + 1
#
# where: f(n) = 4*n*(n+1).

# The sequence a(n) can be factorized as:
#
#   a(n) = (x - y - z + 1)(x + y + z + 1)
#
# where: x = 2^f(n)
#        y = 2^(f(n)/2)
#        z = n * 2^(f(n)/2 + 1)

func f(n) { 4 * n * (n+1) }

func a(n) {
    (2**f(n) - f(n) + 1) * 2**f(n) + 1
}

func factors_of_a(n) {

    var x = 2**f(n)
    var y = 2**(f(n)/2)
    var z = n*(2**(f(n)/2 + 1))

    [(x - y - z + 1), (x + y + z + 1)]
}

for n in (1..5) {
    var u = a(n)
    var t = factors_of_a(n)
    say "#{u} = #{t.join(' * ')}"
    assert_eq(u, t.prod)
}

__END__
63745 = 209 * 305
281474590834689 = 16756737 * 16797697
79228162514251108269638549505 = 281474859270145 * 281475094151169
1461501637330902918203589327576533463951130755073 = 1208925819604733570056193 * 1208925819624524779356161
1766847064778384329583297500742918357649352398470630082568566027931615233 = 1329227995784915860221670509605027841 * 1329227995784915885585943610955661313