#65 - Convergents of $e$
The square root of \(2\) can be written as an infinite continued fraction.
\[\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}\]
The infinite continued fraction can be written, \(\sqrt{2} = [1; (2)]\), \((2)\) indicates that \(2\) repeats ad infinitum. In a similar way, \(\sqrt{23} = [4; (1, 3, 1, 8)]\).
It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for \(\sqrt{2}\).
\[\begin{align} &1 + \dfrac{1}{2} &= \dfrac{3}{2} \\ &1 + \dfrac{1}{2 + \dfrac{1}{2}} &= \dfrac{7}{5}\\ &1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}} &= \dfrac{17}{12}\\ &1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}}} &= \dfrac{41}{29} \end{align}\]
Hence the sequence of the first ten convergents for \(\sqrt{2}\) are:
\[1, \dfrac{3}{2}, \dfrac{7}{5}, \dfrac{17}{12}, \dfrac{41}{29}, \dfrac{99}{70}, \dfrac{239}{169}, \dfrac{577}{408}, \dfrac{1393}{985}, \dfrac{3363}{2378}, ...\]
What is most surprising is that the important mathematical constant,
\[e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ... , 1, 2k, 1, ...]\]
The first ten terms in the sequence of convergents for \(e\) are:
\[2, 3, \dfrac{8}{3}, \dfrac{11}{4}, \dfrac{19}{7}, \dfrac{87}{32}, \dfrac{106}{39}, \dfrac{193}{71}, \dfrac{1264}{465}, \dfrac{1457}{536}, ...\]
The sum of digits in the numerator of the \(10\)th convergent is \(1 + 4 + 5 + 7 = 17\).
Find the sum of digits in the numerator of the \(100\)th convergent of the continued fraction for \(e\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=65. Published Friday, 12th March 2004, 06:00 pm. Solved by 33,798 members at time of mirroring.
Why this is useful
Numerical Computing. Precision, conditioning, and numerical iteration transfer directly to pricing engines and model calibration (Phases 5, 13, 15).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 1.1 Sets, Functions, and Relations · 19.12 Continued Fractions, Pell Equations, and Diophantine Approximation · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 2.1 Functions, Limits, and Continuity · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Concepts: algebra geometry number-theory numerical-methods sequences-series
Likely techniques: continued-fractions digit-dp exact-rational
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Understand the problem
- What exactly is the input to problem 65? Is it a bound (the 100th such term), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- 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 the 100th such term impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly the 100th such term?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by the 100th such term 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?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says sequences-series / continued-fractions - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = the 100th such term, 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 (the statement says: "The first ten terms in the sequence of convergents for e are: 2, 3, frac 83, frac 114, frac 197, frac 8732, frac 10639, frac 19371, frac 1264465, frac 1457536,.") - then run it. A surprise here is worth more than an hour of debugging later.
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You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the sequences-series structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the continued-fractions idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound the 100th such term 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 'continued-fractions' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: explore the same concept filter in the Lab
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.