Aviva Trivia

If you separate each of the digits of the number 8166 and form a sequence by summing them in a Fibonacci-like fashion thus: 8 + 1 + 6 + 6 = 21,  1 + 6 + 6 + 21 = 34, 6 + 6 + 21 + 34 = 67, …, after 14 terms you will see the reversal of 8166 appear. Here is the full sequence: 8, 1, 6, 6, 21, 34, 67, 128, 250, 479, 924, 1781, 3434, 6618.

I have dubbed numbers with this property “revrepfigits,” (reverse replicating Fibonacci-like digits) after “repfigit” numbers (defined below). A formal definition of revrepfigits is, Numbers n such that their reversal occurs in a sequence generated by starting with the n digits of a number and continuing the sequence with a number that is the sum of the previous n terms.

Revrepfigits are similar to repfigits, which were introduced by Mike Keith in 1987. Repfigits are numbers having the same property as revrepfigits except the original number appears in the generated sequence instead of its reversal. For example, 3684 is a repfigit since it occurs in the sequence 3, 6, 8, 4, 21, 39, 72, 136, 268, 515, 991, 1910, 3684. The word Fibonacci in the definition comes from the well-known sequence defined as Fn = Fn – 1  + Fn – 2, F0 = 0, F1 = 1, F2  = 1, (where each term is the sum of the previous two) which produces: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

Here are all the currently known revrepfigits: 12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611, (sequence A097060 in Neil Sloane’s Online Encyclopedia of Integer Sequences).

Notice there are no numbers ending with zeros; they aren’t permitted since the zeros would be dropped upon reversal. But terms with internal zeros such as 90711 are allowed. Also, there are only two primes in the above list: 71 and 1593583.

Revrepfigits seem to be rarer than repfigits, and much rarer than primes. For example, less than 106 there are 17 revrepfigits, while there are 34 repfigits (twice as many), and a whopping 78498 primes. It’s still a mystery whether there are infinitely many revrepfigits or repfigits.

The sequence of revrepfigits listed above was found by exhaustive computer search. But there have been numerous techniques developed over the years to speed up the discovery of the original repfigits. Many papers explaining the algorithms used have appeared in the Journal of Recreational Mathematics. A few people have used techniques that involve Diophantine equations to find repfigits; those techniques combined with integer linear programming allowed Daniel Lichtblau to discover this 29-digit repfigit monstrosity in 2004 (currently the largest known):

70267375510207885242218837404.

Concerning revrepfigits, I have made the following two conjectures.
1. They are infinite.
2. They are rarer than repfigits.

I wouldn’t know where to begin to try to prove them.

Now I will close this article with the sequence produced by the largest known revrepfigit, 8422611.

8, 4, 2, 2, 6, 1, 1, 24, 40, 76, 150, 298, 590, 1179, 2357, 4690, 9340, 18604, 37058, 73818, 147046, 292913, 583469, 1162248.