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What number greater than 1 shows up most often in Pascal's Triangle?

           
   
    While 1 shows up twice in every row of Pascal’s Triangle, every number n shows up at least twice in row n at  ; some numbers show up on at least two rows, like 10, which is  .  Showing up on three rows is pretty good, and there is a formula involving the Fibonacci numbers that can give us an infinite number of numbers that show up on three rows.  Here are the first few Fibonaccis, written alternately in red and blue.

    0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

    The formula that works is n = F_2kF_2k+1 and m = F_2k-1F_2k; this means take any blue Fibonacci number and multiply it by the next bigger red Fibonacci and by the previous smaller red Fibonacci.  Doing so gives us a three row participant r where .  The first interesting number we can get this way is n=3×5=15 and m=2×3=6, so r = 3003, which is ; the next n and m are n=8×13=104 and m=5×8=40, which gives us the number r=61218182743304701891431482520.  While this formula does work, it gives us gigantic numbers to deal with, and there are a lot of numbers that show up three times that aren’t found by this formula.

    The current champ is 3003 = 3×7×11×13, which shows up on four rows; not only is it on the 14^th, 15^th and 3003^rd rows, it also makes an appearance on the 78^th row as ; there may be some number greater than 1 that shows up as often or even more often, but it hasn’t been found yet, and people have used computers to check all the numbers less than 3.3×10¹⁷.