# Tag Archives: Fibonacci number

## Fibonacci in arithmetic progression

Fibonacci numbers is defined by the recurrence relation: then,   giving us all increasing arithmetic progressions formed of three terms of the Fibonacci sequence For example,     Prove that there are no increasing arithmetic progressions formed of … Continue reading

## Fibonacci numbers | Is F(m*n -1) – (F(n-1))^m always divisible by (F(n))^2 ?

Is      always divisible by          for all     and     any counterexample?

## Fibonacci numbers | gcd …. (Part 2)

The gcd of any two Fibonacci numbers is also a Fibonacci number       ->     for all   ->   ->   if     divides   ,   then     divides            If … Continue reading

## The square of a Fibonacci number

Show that is the square of a Fibonacci number

## Fibonacci numbers | is 5 F^2_{2k} + 4 always a square number?

Is     always a square number? Also, is     always a square number?

## Fibonacci numbers | F(n) < x < F(n+1) < y < F(n+2)

If   then show that     is never a Fibonacci number

## Fibonacci numbers | (F(n) F(n+3))^2 + (2 F(n+1) F(n+2))^2

Is   a square number for all positive integers   ?

## Fibonacci num3ers : a surprising occurrence

To find positive integers     such that                     are all squares   that is,                     Note that   It happens that the first few Fibonacci numbers can be used to … Continue reading

## Sequence (A, B, C, D), (D – A) is a Fibonacci number

Fibonacci numbers are the numbers in the following integer sequence : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, … Continue reading

## Fibonacci number | ceiling(√e^n) for n=0,1,…,8

The first few Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, … Continue reading

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