fix parentheses

This commit is contained in:
filifa 2025-07-20 20:05:01 -04:00
parent dd24aac8bb
commit f8d7a1d825
4 changed files with 4 additions and 4 deletions

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@ -11,7 +11,7 @@
"\n",
"The concept the problem is describing is called a [Farey sequence](https://en.wikipedia.org/wiki/Farey_sequence). The example given in the problem is $F_8$, and we are tasked with finding the numerator of the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
"\n",
"It turns out there is a very simple method for determining this. Whenever you have two neighbors $\\frac{a}{b}$ and $\\frac{c}{d}$ in a Farey sequence, the next term that will appear between them in a subsequent Farey sequence is simply $\\frac{a+c}{b+d}$, called the [mediant](https://en.wikipedia.org/wiki/Mediant_(mathematics)) of the two neighbors. For example, since we're given that the left neighbor of $\\frac{3}{7}$ in $F_8$ is $\\frac{2}{5}$, the next fraction to appear between the two will be\n",
"It turns out there is a very simple method for determining this. Whenever you have two neighbors $\\frac{a}{b}$ and $\\frac{c}{d}$ in a Farey sequence, the next term that will appear between them in a subsequent Farey sequence is simply $\\frac{a+c}{b+d}$, called the [mediant](https://w.wiki/EoNc) of the two neighbors. For example, since we're given that the left neighbor of $\\frac{3}{7}$ in $F_8$ is $\\frac{2}{5}$, the next fraction to appear between the two will be\n",
"$$\\frac{2+3}{5+7} = \\frac{5}{12}$$\n",
"Naturally, this fraction will first appear in $F_{12}$, meaning $\\frac{5}{12}$ is the left neighbor of $\\frac{3}{7}$ in that Farey sequence. We could then, in turn, find the mediant of $\\frac{5}{12}$ and $\\frac{3}{7}$ to find the next left neighbor of $\\frac{3}{7}$ ($\\frac{8}{19}$, appearing in $F_{19}$).\n",
"\n",

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"source": [
"# [Counting Summations](https://projecteuler.net/problem=76)\n",
"\n",
"We want $p(100) - 1$, where $p(n)$ is the [partition function](https://en.wikipedia.org/wiki/Partition_function_(number_theory)). We subtract 1 because $p(n)$ counts $n$ by itself as a partition of $n$, but we only want partitions composed of two or more numbers.\n",
"We want $p(100) - 1$, where $p(n)$ is the [partition function](https://w.wiki/EoNj). We subtract 1 because $p(n)$ counts $n$ by itself as a partition of $n$, but we only want partitions composed of two or more numbers.\n",
"\n",
"Guess what? SageMath has this built-in."
]

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@ -44,7 +44,7 @@
"source": [
"Theoretically, we could create a generating function like we did in [problem 31](https://projecteuler.net/problem=76) or [problem 76](https://projecteuler.net/problem=76) to solve this, and could even use $\\mathbb{Z}_{1000000}$ as our base ring to handle the modulus automagically, but since the answer is pretty large, it would be impractical to construct the function with the required precision.\n",
"\n",
"Instead, if you'd like to implement the [partition function](https://en.wikipedia.org/wiki/Partition_function_(number_theory)) yourself, Euler's [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem) leads to a useful recurrence equation:\n",
"Instead, if you'd like to implement the [partition function](https://w.wiki/EoNj) yourself, Euler's [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem) leads to a useful recurrence equation:\n",
"$$p(n) = p(n - 1) + p(n - 2) - p(n - 5) - p(n - 7) + p(n - 12) + p(n - 15) - p(n - 22) - p(n - 26) + \\cdots$$\n",
"Here, the numbers are the [generalized pentagonal numbers](https://en.wikipedia.org/wiki/Pentagonal_number). As base cases, $p(0) = 1$ and $p(n) = 0$ for negative $n$, so the infinite series eventually converges.\n",
"\n",

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"id": "96be7f65",
"metadata": {},
"source": [
"Now with all of the adjustments for special rules in place, we can solve the problem by finding the chain's stationary distribution, which is an [eigenvector](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the stochastic matrix with eigenvalue 1. To find this eigenvector, we'll first find the basis vector of the [kernel](https://en.wikipedia.org/wiki/Kernel_(linear_algebra)) of $Q - I$ (where $I$ is the identity matrix), with the help of SageMath."
"Now with all of the adjustments for special rules in place, we can solve the problem by finding the chain's stationary distribution, which is an [eigenvector](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the stochastic matrix with eigenvalue 1. To find this eigenvector, we'll first find the basis vector of the [kernel](https://w.wiki/EoP5) of $Q - I$ (where $I$ is the identity matrix), with the help of SageMath."
]
},
{