Unreachable Numbers
Consider the equation $17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e. $a,b,c,p \gt 0$. For a given $p$ there are some values...
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John Rhodes
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August 16, 2023
Summation of a Modular Formula
For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$ For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ and so $f(3) = 5$. Further...
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John Rhodes
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August 16, 2023
Sextuplet Norms
Let $f(n)$ be the number of $6$-tuples $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that: All $x_i$ are integers with $0 \leq x_i < n$ $\gcd(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,\ n^2)=1$ Let $\displaystyle G(n)=\displaystyle\sum_{k=1}^n \frac{f(k)}{k^2\varphi(k)}$ where...
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John Rhodes
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August 16, 2023
Turán’s Water Heating System
Turan has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one...
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John Rhodes
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August 16, 2023
Exponent Difference
For any integer $n>0$ and prime number $p,$ define $\nu_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$. Define $$D(n, m) = \sum_{p \text{ prime}}...
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John Rhodes
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August 16, 2023
Binary Blackboard
Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They...
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John Rhodes
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August 16, 2023
One Million Members
On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you...
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John Rhodes
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August 16, 2023
Even Stevens
Every day for the past $n$ days Even Stevens brings home his groceries in a plastic bag. He stores these plastic bags in a cupboard. He either...