What is the 23rd Fibonacci number?

F(23) = 28,657. It comes from adding the two terms before it: F(22) + F(21) = 17,711 + 10,946 = 28,657. It has 5 digits, and the terms either side are 17,711 and 46,368.

Is 28,657 a prime number?

Yes — 28,657 is a Fibonacci prime. Its index 23 is also prime, which is necessary for any Fibonacci prime above F(4). Whether the sequence contains infinitely many primes is still unproved.

What is F(23) divided by F(22)?

28,657 ÷ 17,711 = 1.618033990. The golden ratio is φ = 1.618033989, so this is off by 1.43e-9 — above it, since the ratios alternate on either side as they close in.

What are the Lucas number and running total at index 23?

L(23) = 64,079, which is F(22) + F(24) = 17,711 + 46,368. The running total F(1) + … + F(23) is 75,024, which equals F(25) − 1 = 75,025 − 1 — an identity that holds at every index, not just this one. The squares up to here sum to F(23) × F(24) = 1,328,767,776.

Are there infinitely many Fibonacci primes?

Nobody knows, and it is one of the more embarrassing open problems in the subject given how elementary it sounds. Fibonacci primes are known — 2, 3, 5, 13, 89, 233, 1597 and so on — and very large ones have been found, but no proof exists that the list never ends. One thing is settled: apart from F(4) = 3, a Fibonacci prime can only occur at a prime index, though the reverse fails, as F(19) = 4,181 = 37 × 113 shows.

What is the Zeckendorf representation?

Every positive whole number is a sum of Fibonacci numbers with no two of them adjacent in the sequence, and there is exactly one such sum. For 100 it is 89 + 8 + 3. The greedy method — repeatedly take the largest that fits — always lands on this unique representation, which is a stronger guarantee than greedy algorithms usually give.

How can I check whether a number is in the sequence?

There is an exact test: N is a Fibonacci number if and only if 5N² + 4 or 5N² − 4 is a perfect square. Nothing outside the sequence passes and nothing inside it fails. It beats comparing against φⁿ ÷ √5 because it stays in whole numbers, so it does not degrade for large values the way a floating-point comparison would.

What is the sum of the first n Fibonacci numbers?

F(n+2) − 1, always. Add the first ten and you get 143, which is 144 − 1. The identity is easy to see once stated: each term is a difference of two later ones, so the sum collapses. The sum of the squares is just as tidy — F(1)² + … + F(n)² = F(n) × F(n+1) — which is the algebraic statement of the square tiling that builds the spiral.

Is the Fibonacci spiral the golden spiral?

No. The familiar picture is built from quarter circles, one per square, each of constant radius, so its curvature jumps at every join. A true golden spiral is a smooth logarithmic curve whose radius grows by a factor of φ every quarter turn. They look nearly identical because the Fibonacci ratios approach φ so quickly, and the approximation improves with every square, but they are different curves.

The 23rd Fibonacci Number

The 23rd Fibonacci number is 28,657. Index 23 is a prime term at a prime index, and the last such pairing below index 29. It follows from the rule the whole sequence runs on: add the two terms before it, so F(23) = F(22) + F(21) = 17,711 + 10,946 = 28,657. Divided by the term before it, 28,657 ÷ 17,...

BY POSITION

INDEX n

WORTH TRYING

F(23)

28,657

RATIO TO F(22)

1.618034

5 digits · previous 17,711 · next 46,368 · φ is 1.618034, so this is off by 1.4e-9

Digits

Lucas L(n)

64,079

Sum of F(1)…F(n)

75,024

Ratio to previous

1.618033990

Exact in a double?

yes

Index n

AROUND F(23)

F(n)

DIGITS

6,765

10,946

17,711

28,657

46,368

75,025

121,393

STEPS

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

F(23) = F(22) + F(21) = 17,711 + 10,946

5 digits, small enough to hold exactly in ordinary arithmetic

F(23) = 28,657

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

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HOW TO USE

1
The answer is already here: F(23) = 28,657. This page opens with index 23 loaded, so nothing needs typing.
2
The STEPS panel shows where it comes from: F(22) + F(21) = 17,711 + 10,946 = 28,657.
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

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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.