Determine all possible list(s) of pythagorean triple (a,b,c), with 0 < a < b < c < 100, such that we will obtain another pythagorean triple (p,q,r) by inserting the same nonzero digit to the left of each of a, b and c. None of a, b and c can contain any leading zeroes.

Note: Try to derive a non computer assisted method, although computer program/spreadsheet solutions are welcome.

As the poser has requested all possible list(s) of pythagorean triples (a,b,c) where another pythagorean triple might be obtained by inserting the same nonzero digit to the left of a, b, and c. I take the liberty to interpret 0 < a < b < c < 100 as the limits according to whatever base the pythagorean triple may be represented, i.e., 0_x < a_x < b_x < c_x < 100_x.
Thus, I present the following as a list of solutions for bases up to base-36:

1 & (5₁₀, 12₁₀, 13₁₀) => (15₁₀, 112₁₀, 113₁₀)
1 & (7₁₄, 1A₁₄, 1B₁₄) => (17₁₄, 11A₁₄, 11B₁₄)
1 & (9₁₈, 24₁₈, 25₁₈) => (19₁₈, 124₁₈, 125₁₈)
1 & (B₂₂, 2G₂₂, 2H₂₂) => (1B₂₂, 12G₂₂, 12H₂₂)
1 & (D₂₆, 36₂₆, 37₂₆) => (1D₂₆, 136₂₆, 137₂₆)
1 & (F₃₀, 3N₃₀, 3O₃₀) => (1F₃₀, 13N₃₀, 13O₃₀)
1 & (H₃₄, 48₃₄, 49₃₄) => (1H₃₄, 148₃₄, 149₃₄)
3 & (C₂₄, 1B₂₄, 1D₂₄) => (3C₂₄, 31B₂₄, 31D₂₄)
3 & (G₃₂, 1V₃₂, 21₃₂) => (3G₃₂, 31V₃₂, 321₃₂)

Posted by Dej Mar on 2008-11-10 08:27:34