#25 - $1000$-digit Fibonacci Number
The Fibonacci sequence is defined by the recurrence relation:
\(F_n = F_{n - 1} + F_{n - 2}\), where \(F_1 = 1\) and \(F_2 = 1\).
Hence the first \(12\) terms will be:
\[\begin{align} F_1 &= 1\\ F_2 &= 1\\ F_3 &= 2\\ F_4 &= 3\\ F_5 &= 5\\ F_6 &= 8\\ F_7 &= 13\\ F_8 &= 21\\ F_9 &= 34\\ F_{10} &= 55\\ F_{11} &= 89\\ F_{12} &= 144 \end{align}\]The \(12\)th term, \(F_{12}\), is the first term to contain three digits.
What is the index of the first term in the Fibonacci sequence to contain \(1000\) digits?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=25. Published Friday, 30th August 2002, 06:00 pm. Solved by 169,329 members at time of mirroring.
Why this is useful
Numerical Computing. Matrix methods and linear-recurrence acceleration are the machinery behind covariance work, PCA, and lattice/transition models (Phases 4, 5, 8).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.2 Primes, Sieves, and Integer Factorization · 2.7 Sequences, Series, Convergence, and Power Series
Concepts: dynamic-programming number-theory sequences-series
Likely techniques: matrix-exponentiation memoization
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- What exactly is the input to problem 25? Is it a bound (1000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: one extremal object (or the value attached to it), so a search-with-pruning shape is natural.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What constraint does the bound 1000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000 and the cost of testing one.
- Which sequences-series fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says sequences-series / matrix-exponentiation - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it - then run it. A surprise here is worth more than an hour of debugging later.
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Which step of your solution were you least confident about, and what evidence would settle it?
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Where did the sequences-series structure do the real work? Name the single observation that collapsed the search space.
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How would your solution change if the bound 1000 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'matrix-exponentiation' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
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