Hello Math-Fun, Hans Haverman has done a great job so far (work in progress): (...) [Hans]: We've already decided that the initial digit should not be "0". Let's add the proviso that the initial TWO digits should not be "+0". In addition, let us disallow ANY of the following substrings: {"++", "+00", "+01", "+02", "+03", "+04", "+05", "+06", "+07", "+08", "+09", "=00", "=01", "=02", "=03", "=04", "=05", "=06", "=07", "=08", "=09", "=+0"}. There might be other things we won't like, but let's see what we get so far... (...) Number of 6-digit solutions: =\+ 0 1 2 3 4 5 6 7 8 1 0 2 54 0 3 52 0 48 4 50 0 46 44 5 48 0 46 44 42 6 48 0 42 42 44 44 7 46 0 42 44 40 42 40 8 44 0 44 42 40 40 36 36 9 42 0 42 40 38 38 36 34 30 We see that coding '=' with 2 and '+' with 0 produces 54 terms (54 is the highest integer in the above array); this gives the longest finite sequence of 6-digit integers with such a substitution: S(6) = 112308, 112407, 112506, 112605, 112704, 112803, 132409, 132508, 132607, 132706, 132805, 132904, 142509, 142608, 142707, 142806, 142905, 152609, 152708, 152807, 152906, 162709, 162808, 162907, 172809, 172908, 182909, 308211, 407211, 409213, 506211, 508213, 509214, 605211, 607213, 608214, 609215, 704211, 706213, 707214, 708215, 709216, 803211, 805213, 806214, 807215, 808216, 809217, 904213, 905214, 906215, 907216, 908217, 909218. --- Full explanation and more (5-digit and 7-digit integer) here: http://www.cetteadressecomportecinquantesignes.com/DigitSubstitution.htm 8-digit results to come. Best, É.