##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Arthur Benjamin

#### Harvey Mudd College

## Power

##### Abstract:

When does one Fibonacci number divide another? Let $F_0 = 0$, $F_1 = 1$, and for $n\\geq 2$, $F_n = F_{n-1} + F_{n-2}$. It is well known that for $F_m > 1$ This last result was used in Yuri Matijasevi\\u{c}\'s solution of Hilbert\'s 10th problem. Using simple combinatorial arguments, we derive previuosly unknown necessary and sufficient conditions for the following question: For any $L \\geq 1$ When does $F_m^L$ divide $F_n$? Our method allows us to answer this same question for any Lucas sequence of the first kind, defined by $U_0 = 0$, $U_1 = 1$, and for $n\\geq 2$, $U_n = aU_{n-1} + bU_{n-2}$. This talk is based on joint work with Harvey Mudd College undergraduate Jeremy Rouse, while attending the 10th International Conference on Applications of Fibonacci Numbers.

Host: Fan Chung Graham

### May 27, 2003

### 4:00 PM

### AP&M 7321

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