What is the 15th Fibonacci number?

F(15) = 610. It comes from adding the two terms before it: F(14) + F(13) = 377 + 233 = 610. It has 3 digits, and the terms either side are 377 and 987.

Is 610 a prime number?

No — 610 is composite. It has to be: F(3) = 2 and F(5) = 5 divide it, because 3 and 5 divide 15.

What is F(15) divided by F(14)?

610 ÷ 377 = 1.618037135. The golden ratio is φ = 1.618033989, so this is off by 3.15e-6 — above it, since the ratios alternate on either side as they close in.

What are the Lucas number and running total at index 15?

L(15) = 1,364, which is F(14) + F(16) = 377 + 987. The running total F(1) + … + F(15) is 1,596, which equals F(17) − 1 = 1,597 − 1 — an identity that holds at every index, not just this one. The squares up to here sum to F(15) × F(16) = 602,070.

Where is the sequence actually used?

In more places than the folklore suggests, and mostly unglamorous ones. Fibonacci heaps give the fastest known asymptotic bounds for several graph algorithms. Fibonacci search divides a sorted array using these numbers rather than halves. Fibonacci coding produces a self-synchronising variable-length encoding, and the Zeckendorf representation is what makes it work. Agile teams estimate in Fibonacci numbers because the widening gaps discourage false precision.

What happens at negative indices?

The rule runs backwards perfectly well. Rearranging F(n) = F(n−1) + F(n−2) gives F(n−2) = F(n) − F(n−1), which extends the sequence to negative indices: F(−1) = 1, F(−2) = −1, F(−3) = 2, F(−4) = −3. The pattern is F(−n) = (−1)ⁿ⁺¹ F(n), so the magnitudes mirror the positive side and the signs alternate. This page covers the non-negative side.

Why do the ratios alternate around φ?

Because of the second root. Binet’s formula writes F(n) as a combination of φⁿ and ψⁿ, where ψ = (1 − √5) ÷ 2 is about −0.618. Since ψ is negative and smaller than 1 in size, its powers shrink towards zero while flipping sign at every step — and that flipping is exactly what pushes each successive ratio to the other side of φ. It also explains the speed: the error is governed by ψⁿ, which decays geometrically.

What is the Fibonacci sequence?

A sequence in which each term is the sum of the two before it, beginning 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. It reached European mathematics through Leonardo of Pisa in 1202, as a puzzle about breeding rabbits, though it appears in Indian prosody several centuries earlier. The rabbit setup is artificial; the sequence turned out to matter for entirely different reasons.

How is the golden ratio involved?

Divide any term by the one before it and the answer approaches φ = 1.618033988749895, alternating above and below as it goes. The error shrinks by roughly a factor of φ² at each step, so convergence is quick: by the fortieth term the ratio matches φ to every digit an ordinary computer number can hold. Any sequence built by adding the last two terms does this, whatever it starts from.

The 15th Fibonacci Number

The 15th Fibonacci number is 610. Index 15 is a term whose value is divisible by both 5 and 2, since F(5) and F(3) both divide it. The rule is exact: F(d) divides F(n) whenever d divides n. It follows from the rule the whole sequence runs on: add the two terms before it, so F(15) = F(14) + F(13) = 3...

BY POSITION

INDEX n

WORTH TRYING

F(15)

610

RATIO TO F(14)

1.618037

3 digits · previous 377 · next 987 · φ is 1.618034, so this is off by 3.1e-6

Digits

Lucas L(n)

1,364

Sum of F(1)…F(n)

1,596

Ratio to previous

1.618037135

Exact in a double?

yes

Index n

AROUND F(15)

F(n)

DIGITS

144

233

377

610

987

1,597

2,584

STEPS

Every term is the sum of the two before it, starting from F(0) = 0 and F(1) = 1

F(15) = F(14) + F(13) = 377 + 233

3 digits, small enough to hold exactly in ordinary arithmetic

F(15) = 610

GOLDEN SPIRAL

The squares build the sequence. Each new square takes the whole long side of what is already there, so its side is the sum of the two before it — which is the Fibonacci rule stated in geometry rather than arithmetic. The rectangle is always F(k) by F(k+1), and the quarter arcs join end to end into the spiral. It approaches the golden spiral without ever being one: this is built from circular arcs, and the true golden spiral is a smooth logarithmic curve.

THE RATIO CLOSING ON φ

n = 3

2.00000000

n = 5

1.66666667

n = 8

1.61538462

n = 12

1.61797753

n = 20

1.61803396

n = 40

1.61803399

Bar length is the distance from φ = 1.618033989, so shorter is closer. The ratio alternates above and below it and the error falls by roughly a factor of φ² each step, which is why n = 40 already matches φ to the last digit a double can hold.

Exact, not floating point. F(78) is the last term that fits exactly in ordinary double arithmetic; F(79) is 14,472,334,024,676,221, which is past the limit. Everything here is computed with exact big integers, so F(500) is right to the final digit rather than rounded. Indices run to 1,000.

Live simulation · drawn to scale · updates as you type

Created with❤️byeaglecalculator.com

HOW TO USE

1
The answer is already here: F(15) = 610. This page opens with index 15 loaded, so nothing needs typing.
2
The STEPS panel shows where it comes from: F(14) + F(13) = 377 + 233 = 610.
3
The six tiles carry the derived quantities — digit count, the Lucas number, the running total, the ratio to the previous term, and whether the value still fits exactly in ordinary computer arithmetic.
4
Change the index to carry on, or switch to IS IT ONE? to test whether some other number appears in the sequence at all.

THE FORMULAS

The ruleF(n) = F(n−1) + F(n−2)

The seedsF(0) = 0, F(1) = 1

Golden ratioφ = (1 + √5) ÷ 2

Binet's formulaF(n) = (φⁿ − ψⁿ) ÷ √5

Ratio limitF(n) ÷ F(n−1) → φ

Sum of the first nF(1) + … + F(n) = F(n+2) − 1

Sum of squaresF(1)² + … + F(n)² = F(n)·F(n+1)

Cassini identityF(n)² − F(n−1)F(n+1) = (−1)ⁿ⁻¹

Greatest common divisorgcd(F(m), F(n)) = F(gcd(m,n))

Lucas numbersL(n) = F(n−1) + F(n+1)

Fibonacci test5N² ± 4 is a perfect square

Zeckendorfa unique sum of non-consecutive terms

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

What is the 7th Fibonacci Number?

Calculate instantly →

What is the 39th Fibonacci Number?

Calculate instantly →

What is the 40th Fibonacci Number?

Calculate instantly →

What is the 16th Fibonacci Number?

Calculate instantly →

MORE SEQUENCES CALCULATORS

What is the 3rd Fibonacci Number?

Calculate instantly →

What is the 4th Fibonacci Number?

Calculate instantly →

What is the 5th Fibonacci Number?

Calculate instantly →

What is the 6th Fibonacci Number?

Calculate instantly →

What is the 7th Fibonacci Number?

Calculate instantly →

What is the 8th Fibonacci Number?

Calculate instantly →

What is the 9th Fibonacci Number?

Calculate instantly →

What is the 10th Fibonacci Number?

Calculate instantly →

What is the 11th Fibonacci Number?

Calculate instantly →

What is the 12th Fibonacci Number?

Calculate instantly →

What is the 13th Fibonacci Number?

Calculate instantly →

What is the 14th Fibonacci Number?

Calculate instantly →

What is the 16th Fibonacci Number?

Calculate instantly →

What is the 1st Fibonacci Number?

Calculate instantly →

Fibonacci Calculator

Calculate instantly →

What is the 17th Fibonacci Number?

Calculate instantly →

What is the 18th Fibonacci Number?

Calculate instantly →

What is the 19th Fibonacci Number?

Calculate instantly →

What is the 2nd Fibonacci Number?

Calculate instantly →

Was this calculator helpful?

Last updated: July 29, 2026 · Formula verified · Exact big-integer arithmetic, so terms past F(78) are right to the final digit · Eagle-eyed accuracy for every calculation.