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- #!/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
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