Pisano periods
Does the Fibonacci sequence repeat?
The digits do, proven
The numbers never repeat, they grow forever. Their digits always do: the last digit repeats every 60 terms, the last two digits every 300, and the digital roots every 24. It is proven for any number you divide by, because each remainder depends only on the two before it, and there are only so many pairs. Joseph Louis Lagrange noticed the 60 in 1774.
Divide by
Repeats every
…
The reason
Only so many pairs
Each Fibonacci number is the sum of the two before: 0, 1, 1, 2, 3, 5, 8, 13, 21... Keep only the remainder after dividing by some number m, and the same rule holds for the remainders: each one is the sum of the last two, divided by m.
So the whole future of the sequence is fixed by the latest pair. There are only m × m possible pairs, so sooner or later a pair must come round again, and from that moment everything repeats. The rule also runs backwards, so the first pair to come back is always the starting pair, 0 and 1. That repeat length is called the Pisano period, after Leonardo Pisano, the man we call Fibonacci.
Dividing by 10 keeps the last digit, and the period is 60. Dividing by 100 keeps the last two digits, and the period is 300. Dividing by 9 gives the digital roots, the digit sums taken down to one digit, and the period is 24. None of this depends on base 10: every modulus repeats.
The 24 cycle
The pattern numerology loves
The digital roots of the Fibonacci numbers, starting from the first 1, run:
1 1 2 3 5 8 4 3 7 1 8 9 · 8 8 7 6 4 1 5 6 2 8 1 9
and then start again. The second half mirrors the first: add any root to the one twelve places later and you get 9, 1 + 8, 1 + 8, 2 + 7 and so on. The exception is the twelfth place of each half, where both are 9. The exact rule is that F(n + 12) and −F(n) leave the same remainder after dividing by 9.
It is real, and it is the kind of pattern vortex math reads as cosmic. But it is one Pisano period among infinitely many. Count in base 12 and digital roots work on 11 instead of 9, and the cycle is 10 long. Watch it change on the base switch.
History and nature
India first, sunflowers yes, nautilus no
Fibonacci did not discover it. Indian writers on poetry counted rhythm patterns with these numbers long before him. The sequence appears in the work of Pingala as early as 200 BC. Virahanka described it around 700, and Gopala (before 1135) and Hemachandra (around 1150) stated it plainly: each number is the sum of the last two. Leonardo of Pisa brought it to Europe in his Liber Abaci of 1202 with a puzzle about breeding rabbits. The name “Fibonacci sequence” came from Édouard Lucas in the 19th century.
Plants really do use it. Leaves around a stem turn by fractions such as 2/5 in oak, 3/8 in sunflowers and 5/13 in willow, and sunflower heads show spiral counts that are Fibonacci numbers. The Bravais brothers linked leaf arrangement to the sequence in 1837.
The nautilus does not. Its shell is a logarithmic spiral, but measurements do not support the claim that it is a golden spiral. The mathematician Clement Falbo wrote in 2005 that the spirals of the Nautilus pompilius “are not in the shape of the golden ratio, as is often claimed.”
Real vs legend
Fibonacci, sorted
| Claim | Status |
|---|---|
| Fibonacci digits repeat: 60, 300 and 24 terms. | Real · Proven |
| Digital roots 12 apart add to 9. | Almost · 9 and 9 at every 12th |
| Ratios of neighbours approach the golden ratio, 1.618... | Real |
| Plants grow in Fibonacci spirals. | Real · leaves and seed heads |
| Fibonacci discovered the sequence. | No · India, centuries earlier |
| The nautilus shell is a golden spiral. | Legend |
| 432 is a Fibonacci number. | No · 377, then 610 |
Questions
The Fibonacci sequence
Does the Fibonacci sequence repeat?
The numbers themselves never repeat, they keep growing. Their remainders always repeat: the last digit every 60 terms, the last two digits every 300, and the digital roots every 24. This is proven for every number you divide by.
What is so magical about Fibonacci numbers?
Nothing supernatural, but a lot that is neat. Each is the sum of the two before, the ratio of neighbours closes in on the golden ratio 1.618..., their digits repeat in cycles, and they turn up in the way many plants arrange leaves and seeds.
Why is the Fibonacci sequence so important?
It was first used to count real things, the rhythm patterns of Sanskrit poetry. It describes leaf and seed arrangements in plants, and it is tied to the golden ratio by an exact formula.
Is 432 a Fibonacci number?
No. The sequence goes 233, 377, 610, so it jumps straight past 432.
Who discovered the Fibonacci sequence first?
Indian scholars of poetry, from Pingala around 200 BC to Virahanka around 700 and Gopala and Hemachandra in the 1100s. Leonardo of Pisa, later called Fibonacci, introduced it to Europe in 1202.
Is the nautilus shell a Fibonacci spiral?
No. It is a logarithmic spiral, one of the finest in nature, but measurements of real shells do not match the golden spiral that the Fibonacci numbers approach.
Sources
Where this comes from
- Pisano periods and Lagrange, 1774: Wikipedia. OEIS A001175.
- Digital roots of the Fibonacci numbers: OEIS A030132.
- Fibonacci sequence, its Indian history and the golden ratio: Wikipedia.
- The Fibonacci numbers: OEIS A000045.
- Phyllotaxis, leaf fractions and the Bravais brothers: Wikipedia.
- Golden ratio and the nautilus claim: Wikipedia. Nautilus: Wikipedia.
- Clement Falbo, “The Golden Ratio: A Contrary Viewpoint,” College Mathematics Journal, 2005: ERIC.
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The 24 digit Fibonacci cycle, printed and framed.