legendary_question_six.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 29 June 2019
# https://github.com/trizen

# Efficient method for computing non-trivial solutions to the legendary question six.

# The Legend of Question Six - Numberphile
# https://www.youtube.com/watch?v=Y30VF3cSIYQ

# The Return of the Legend of Question Six - Numberphile
# https://www.youtube.com/watch?v=L0Vj_7Y2-xY

# Solutions for (a^2 + b^2) / (1 + ab) = 4, are given by A052530 = { 2, 8, 30, 112, 418, 1560, 5822, ... }.

# Example:
#   (  2^2 +   8^2) / (    2*8 + 1) = 4
#   (  8^2 +  30^2) / (   8*30 + 1) = 4
#   ( 30^2 + 112^2) / ( 30*112 + 1) = 4
#   (112^2 + 418^2) / (112*418 + 1) = 4

# Similar sequences provide solutions for other values:

# For 3^2: A065100 = { 3, 27, 240, 2133, 18957, 168480, ... }
# For 4^2: A154021 = { 4, 64, 1020, 16256, 259076, 4128960, ... }
# For 5^2: A154022 = { 5, 125, 3120, 77875, 1943755, 48516000, ... }
# For 6^2: A154023 = { 6, 216, 7770, 279504, 10054374, 361677960, ... }

# More generally, let U(n,x) be the Chebyshev polynomials of the second kind, then (a^2 + b^2) / (1 + ab) = m^2, has solutions of the form:
#
#   a = m * U(n, m^2 / 2)
#   b = m * U(n+1, m^2 / 2)
#
# for any given positive integer m.

func lq6_solutions(m, how_many=5) {
    how_many.of {|n|
        [m * chebyshevU(n, m*m / 2), m * chebyshevU(n+1, m*m / 2)]
    }
}

for k in (2..10) {
    var S = lq6_solutions(k)
    say "(a^2 + b^2) / (1 + ab) = #{k}^2 has solutions: #{S}"
}

__END__
(a^2 + b^2) / (1 + ab) = 2^2 has solutions: [[2, 8], [8, 30], [30, 112], [112, 418], [418, 1560]]
(a^2 + b^2) / (1 + ab) = 3^2 has solutions: [[3, 27], [27, 240], [240, 2133], [2133, 18957], [18957, 168480]]
(a^2 + b^2) / (1 + ab) = 4^2 has solutions: [[4, 64], [64, 1020], [1020, 16256], [16256, 259076], [259076, 4128960]]
(a^2 + b^2) / (1 + ab) = 5^2 has solutions: [[5, 125], [125, 3120], [3120, 77875], [77875, 1943755], [1943755, 48516000]]
(a^2 + b^2) / (1 + ab) = 6^2 has solutions: [[6, 216], [216, 7770], [7770, 279504], [279504, 10054374], [10054374, 361677960]]
(a^2 + b^2) / (1 + ab) = 7^2 has solutions: [[7, 343], [343, 16800], [16800, 822857], [822857, 40303193], [40303193, 1974033600]]
(a^2 + b^2) / (1 + ab) = 8^2 has solutions: [[8, 512], [512, 32760], [32760, 2096128], [2096128, 134119432], [134119432, 8581547520]]
(a^2 + b^2) / (1 + ab) = 9^2 has solutions: [[9, 729], [729, 59040], [59040, 4781511], [4781511, 387243351], [387243351, 31361929920]]
(a^2 + b^2) / (1 + ab) = 10^2 has solutions: [[10, 1000], [1000, 99990], [99990, 9998000], [9998000, 999700010], [999700010, 99960003000]]