#204 - Generalised Hamming Numbers
A Hamming number is a positive number which has no prime factor larger than \(5\).
So the first few Hamming numbers are \(1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15\).
There are \(1105\) Hamming numbers not exceeding \(10^8\).
We will call a positive number a generalised Hamming number of type \(n\), if it has no prime factor larger than \(n\).
Hence the Hamming numbers are the generalised Hamming numbers of type \(5\).
How many generalised Hamming numbers of type \(100\) are there which don't exceed \(10^9\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=204. Published Saturday, 6th September 2008, 02:00 pm. Solved by 8,284 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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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 · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.2 Primes, Sieves, and Integer Factorization
Recommended stepping-stone problems: #32 · #38 · #97
Concepts: number-theory
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- What exactly is the input to problem 204? Is it a bound (10^9), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer.
- 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 10^9 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10^9?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^9 and the cost of testing one.
- Which number-theory 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 complexity of your intended method in terms of N = 10^9, and the wall-clock time you expect. Write both down now.
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- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
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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 number-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the number-theory idea faster? Which words in the statement were pointing at it, and did you notice them?
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How would your solution change if the bound 10^9 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?
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- Related problem: #32 · #38 · #97
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