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Sums of Fibonacci numbers with few binary digits have been of quite some interest lately. For instance, Bugeaud, Cipu and Mignotte [4] determined all Fibonacci numbers with at most four binary digits. Bravo and Luca [3] found all sums of exactly two Fibonacci numbers which are a perfect power of two. Chim and Ziegler [7] resolved the two Diophantine equations (1) Fn1 + Fn2 = 2 a1 + 22 + 23 and Fn1 + Fn2 + Fn3 = 2 a1 + 22 , where Fk denotes the k-th Fibonacci number (defined by F0 = 0, F1 = 1 and Fk+2 = Fk+1 + Fk for k ≥ 0). These problems are special cases of the following more general problem: Find all numbers that have few digits with respect to both the Zeckendorf and the binary representation. The second author proved a result about linear equations in members of two recurrence sequences [15] which implies that such problems have only finitely many solutions. Moreover, he combined the above mentioned results and completely solved the problem for the bound M = 5 of total number of digits. The standard strategy for solving Diophantine equations like (1) is the iterated application of linear forms in logarithms. It results in a huge upper bound for the largest variable. The huge bound is then reduced by the Baker-Davenport reduction method. The challenge is, that the reduction process requires a large number of computations and that the computation time grows exponentially with the number of variables. Using a usual PC and an efficient algorithm, one can solve such equations with up to six variables in a reasonable time. Seven variables would already take about a month’s time. In this paper we show that by using p-adic results one can skip one of the steps of the general procedure and thus solve equations with even more variables. The p-adic results in question are results on lower bounds for linear forms in p-adic logarithms and an idea by Pethő and de Weger [11]. Pink and Ziegler [12] used similar ideas to solve the Diophantine equation Fn + Fm = 2 13253 · · · 19946.

On sums of Fibonacci numbers with few binary digits