#153 - Investigating Gaussian Integers
As we all know the equation \(x^2=-1\) has no solutions for real \(x\).
If we however introduce the imaginary number \(i\) this equation has two solutions: \(x=i\) and \(x=-i\).
If we go a step further the equation \((x-3)^2=-4\) has two complex solutions: \(x=3+2i\) and \(x=3-2i\).
\(x=3+2i\) and \(x=3-2i\) are called each others' complex conjugate.
Numbers of the form \(a+bi\) are called complex numbers.
In general \(a+bi\) and \(a-bi\) are each other's complex conjugate.
A Gaussian Integer is a complex number \(a+bi\) such that both \(a\) and \(b\) are integers.
The regular integers are also Gaussian integers (with \(b=0\)).
To distinguish them from Gaussian integers with \(b \ne 0\) we call such integers "rational integers."
A Gaussian integer \(a+bi\) is called a divisor of a rational integer \(n\) if the result \(\dfrac n {a + bi}\) is also a Gaussian integer.
If for example we divide \(5\) by \(1+2i\) we can simplify \(\dfrac{5}{1 + 2i}\) in the following manner:
Multiply numerator and denominator by the complex conjugate of \(1+2i\): \(1-2i\).
The result is \(\dfrac{5}{1 + 2i} = \dfrac{5}{1 + 2i}\dfrac{1 - 2i}{1 - 2i} = \dfrac{5(1 - 2i)}{1 - (2i)^2} = \dfrac{5(1 - 2i)}{1 - (-4)} = \dfrac{5(1 - 2i)}{5} = 1 - 2i\).
So \(1+2i\) is a divisor of \(5\).
Note that \(1+i\) is not a divisor of \(5\) because \(\dfrac{5}{1 + i} = \dfrac{5}{2} - \dfrac{5}{2}i\).
Note also that if the Gaussian Integer \((a+bi)\) is a divisor of a rational integer \(n\), then its complex conjugate \((a-bi)\) is also a divisor of \(n\).
In fact, \(5\) has six divisors such that the real part is positive: \(\{1, 1 + 2i, 1 - 2i, 2 + i, 2 - i, 5\}\).
The following is a table of all of the divisors for the first five positive rational integers:
| \(n\) | Gaussian integer divisors with positive real part | Sum \(s(n)\) of these divisors |
| \(1\) | \(1\) | \(1\) |
| \(2\) | \(1, 1+i, 1-i, 2\) | \(5\) |
| \(3\) | \(1, 3\) | \(4\) |
| \(4\) | \(1, 1+i, 1-i, 2, 2+2i, 2-2i,4\) | \(13\) |
| \(5\) | \(1, 1+2i, 1-2i, 2+i, 2-i, 5\) | \(12\) |
For divisors with positive real parts, then, we have: \(\sum \limits_{n = 1}^{5} {s(n)} = 35\).
\(\sum \limits_{n = 1}^{10^5} {s(n)} = 17924657155\).
What is \(\sum \limits_{n = 1}^{10^8} {s(n)}\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=153. Published Saturday, 5th May 2007, 10:00 am. Solved by 3,135 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.
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 · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.2 Primes, Sieves, and Integer Factorization · 2.1 Functions, Limits, and Continuity · 4.2 Linear Maps, Matrices, Rank, and the Null Space
Recommended stepping-stone problems: #66 · #243 · #804
Concepts: algebra number-theory brute-force-reduction
Likely techniques: exact-rational factorization
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Understand the problem
- What exactly is the input to problem 153? Is it a bound (10), 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 do the arguments of s(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?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10 and the cost of testing one.
- Which algebra 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 strategy: in one sentence, what will your solution do? (The classification says algebra / factorization - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10, 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: "Note that 1+i is not a divisor of 5 because frac 51 + i = frac 52 - frac 52i.") - 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 algebra 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 10 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: #66 · #243 · #804
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