250 Problems in Elementary Number Theory - Sierpinski (1970)

44 250 PROBLEMS IN NUMBER THEORY(since 1 > 2). It follows that there are no four terms of the Fibonacci sequencewhich form an arithmetic progression.65*. We know [27, p. 279, problem 3] that if m is a positive integer,then the remainders of dividing successive terms of the Fibonacci sequenceby m form a periodic sequence with pure period. For m = 2, 3, 4, 5, 6, 7,the remainders upon dividing the terms of the Fibonacci sequence by mare respectively (we show here only first few remainders, and not all ofthem):for m = 2: 1, 1, 0, ... ,for m = 3: 1, 1, 2, 0, ... ,for m = 4: 1, 1, 2, 3, 1,0, ... ,for m = 5: 1, 1, 2, 3, 0, 3, 3, 1, 4, ... ,for m = 6: 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, ... ,for m = 7: 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, ... .Since for every positive integer m ~ 7 all possible remainders modulo mappear in the above sequences, we see that each of the arithmetic progressionswith the difference m ~ 7 contains infinitely many terms of the Fibonaccisequence.We shall show now that the progression 8k+4 (k = 0, 1, 2, ... ) does notcontain any term of the Fibonacci sequence.Since Ul = U2 = 1 and U,,+2 = U,,+Un+l for n = 1, 2, ... , we easily seethat the numbers Ul, U2, ... , U14 give the following remainders upon dividingby 8:1. 1. 2, 3, 5, 0, 5, 5,2, 7, 1, 0, 1, 1.It follows that 81u13~Ul and 8IuI4-u2. Thus, for n· = 1 we have 8Iu"+12-unand 8Iu"+13-U"+1 .Suppose now that these two formulas hold for some positive integers n.We have then 8Iun+12+U,,+13-(U,,+Un+1) or 8Iu,,:"14-Un+2 (since 8Iun~13-U"+I).It follows by induction that 8Iun+12-u" for n = 1, 2, ... , which shows thatthe sequence of consecutive remainders modulo 8 of the Fibonacci sequenceis periodic and has a pure twelve-term period.From the first fourteen remainders modulo 8 we see that these remaindersmay be only 0, 1, 2, 3, 5, and 7. Thus there are no remainders 4 or 6, whichimplies that the progressions 8k+4 and 8k+6 do not contain any termof the Fibonacci sequence. These are the progressions of integer terms withthe desired property, and with the least possible difference.