#556 - Squarefree Gaussian Integers
A Gaussian integer is a number \(z = a + bi\) where \(a\), \(b\) are integers and \(i^2 = -1\).
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which \(b = 0\).
A Gaussian integer unit is one for which \(a^2 + b^2 = 1\), i.e. one of \(1, i, -1, -i\).
Let's define a proper Gaussian integer as one for which \(a \gt 0\) and \(b \ge 0\).
A Gaussian integer \(z_1 = a_1 + b_1 i\) is said to be divisible by \(z_2 = a_2 + b_2 i\) if \(z_3 = a_3 + b_3 i = z_1 / z_2\) is a Gaussian integer.
\(\frac {z_1} {z_2} = \frac {a_1 + b_1 i} {a_2 + b_2 i} = \frac {(a_1 + b_1 i)(a_2 - b_2 i)} {(a_2 + b_2 i)(a_2 - b_2 i)} = \frac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + \frac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}i = a_3 + b_3 i\)
So, \(z_1\) is divisible by \(z_2\) if \(\frac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2}\) and \(\frac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}\) are integers.
For example, \(2\) is divisible by \(1 + i\) because \(2/(1 + i) = 1 - i\) is a Gaussian integer.
A Gaussian prime is a Gaussian integer that is divisible only by a unit, itself or itself times a unit.
For example, \(1 + 2i\) is a Gaussian prime, because it is only divisible by \(1\), \(i\), \(-1\), \(-i\), \(1 + 2i\), \(i(1 + 2i) = i - 2\), \(-(1 + 2i) = -1 - 2i\) and \(-i(1 + 2i) = 2 - i\).
\(2\) is not a Gaussian prime as it is divisible by \(1 + i\).
A Gaussian integer can be uniquely factored as the product of a unit and proper Gaussian primes.
For example \(2 = -i(1 + i)^2\) and \(1 + 3i = (1 + i)(2 + i)\).
A Gaussian integer is said to be squarefree if its prime factorization does not contain repeated proper Gaussian primes.
So \(2\) is not squarefree over the Gaussian integers, but \(1 + 3i\) is.
Units and Gaussian primes are squarefree by definition.
Let \(f(n)\) be the count of proper squarefree Gaussian integers with \(a^2 + b^2 \le n\).
For example \(f(10) = 7\) because \(1\), \(1 + i\), \(1 + 2i\), \(1 + 3i = (1 + i)(2 + i)\), \(2 + i\), \(3\) and \(3 + i = -i(1 + i)(1 + 2i)\) are squarefree, while \(2 = -i(1 + i)^2\) and \(2 + 2i = -i(1 + i)^3\) are not.
You are given \(f(10^2) = 54\), \(f(10^4) = 5218\) and \(f(10^8) = 52126906\).
Find \(f(10^{14})\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=556. Published Sunday, 17th April 2016, 07:00 am. Solved by 311 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.8 Bit Manipulation and State Compression · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #330 · #672 · #937
Concepts: combinatorics number-theory brute-force-reduction
Likely techniques: bitmask-dp factorization prime-test
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Understand the problem
- What exactly is the input to problem 556? Is it a bound (10^14), 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 f(n), f(10) 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 10^14?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^14 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 / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^14, 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, 2 is divisible by 1 + i because 2/(1 + i) = 1 - i is a Gaussian integer.") - 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 number-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-dp 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 10^14 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 'bitmask-dp' 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: #330 · #672 · #937
- 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.