#652 - Distinct Values of a Proto-logarithmic Function
Consider the values of \(\log_2(8)\), \(\log_4(64)\) and \(\log_3(27)\). All three are equal to \(3\).
Generally, the function \(f(m,n)=\log_m(n)\) over integers \(m,n \ge 2\) has the property that
\(f(m_1,n_1)=f(m_2,n_2)\) if
- \(\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,\) for some integers \(a,b,e,f \, \,\) or
- \( \, m_1=a^e, n_1=b^e, m_2=a^f,n_2=b^f \,\) for some integers \(a,b,e,f \,\)
We call a function \(g(m,n)\) over integers \(m,n \ge 2\) proto-logarithmic if
- \(\quad \, \, \, \, g(m_1,n_1)=g(m_2,n_2)\) if any integers \(a,b,e,f\) fulfilling 1. or 2. can be found
- and \(\, g(m_1,n_1) \ne g(m_2,n_2)\) if no integers \(a,b,e,f\) fulfilling 1. or 2. can be found.
Let \(D(N)\) be the number of distinct values that any proto-logarithmic function \(g(m,n)\) attains over \(2\le m, n\le N\).
For example, \(D(5)=13\), \(D(10)=69\), \(D(100)=9607\) and \(D(10000)=99959605\).
Find \(D(10^{18})\), and give the last \(9\) digits as answer.
Note: According to the four exponentials conjecture the function \(\log_m(n)\) is proto-logarithmic.
While this conjecture is yet unproven in general, \(\log_m(n)\) can be used to calculate \(D(N)\) for small values of \(N\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=652. Published Saturday, 19th January 2019, 10:00 pm. Solved by 199 members at time of mirroring.
Why this is useful
Optimization. The transferable skill is replacing infeasible enumeration with a mathematical reduction - the core move in calibration and large-scale computation (Phases 10, 13).
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Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps
Recommended stepping-stone problems: #466 · #676 · #442
Concepts: brute-force-reduction
Likely techniques: big-integer hashing
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Understand the problem
- What exactly is the input to problem 652? Is it a bound (10^18), 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 do the arguments of g(m,n), D(N) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10^18?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^18 and the cost of testing one.
- Which brute-force-reduction 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 brute-force-reduction / big-integer - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^18, 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: "For example, D(5)=13, D(10)=69, D(100)=9607 and D(10000)=99959605.") - then run it. A surprise here is worth more than an hour of debugging later.
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- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
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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 brute-force-reduction structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the big-integer 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 10^18 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 'big-integer' 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?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #466 · #676 · #442
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