#120 - Square Remainders
Let \(r\) be the remainder when \((a - 1)^n + (a + 1)^n\) is divided by \(a^2\).
For example, if \(a = 7\) and \(n = 3\), then \(r = 42\): \(6^3 + 8^3 = 728 \equiv 42 \mod 49\). And as \(n\) varies, so too will \(r\), but for \(a = 7\) it turns out that \(r_{\mathrm{max}} = 42\).
For \(3 \le a \le 1000\), find \(\sum r_{\mathrm{max}}\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=120. Published Friday, 21st April 2006, 06:00 pm. Solved by 15,654 members at time of mirroring.
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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 · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 3.1 Vectors, Multivariable Functions, and Level Sets
Recommended stepping-stone problems: #7 · #20 · #36
Concepts: geometry number-theory
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Understand the problem
- What exactly is the input to problem 120? Is it a bound (1000), 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 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 number-theory 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 number-theory - 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 (the statement says: "For example, if a = 7 and n = 3, then r = 42: 6^3 + 8^3 = 728 = 42 mod 49.") - 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 number-theory 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?
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State the transferable technique in one sentence, without mentioning this problem's story at all.
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- Related problem: #7 · #20 · #36
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