Goldbach’s Other Conjecture
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. \begin{align} 9 =...
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John Rhodes
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August 15, 2023
Triangular, Pentagonal, and Hexagonal
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: Triangle $T_n=n(n+1)/2$ $1, 3, 6, 10, 15, \dots$ Pentagonal $P_n=n(3n – 1)/2$ ...
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John Rhodes
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August 15, 2023
Pentagon Numbers
Pentagonal numbers are generated by the formula, $P_n=n(3n-1)/2$. The first ten pentagonal numbers are: $$1, 5, 12, 22, 35, 51, 70, 92, 117, 145, \dots$$ It can...
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John Rhodes
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August 15, 2023
Sub-string Divisibility
The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but...
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John Rhodes
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August 15, 2023
Coded Triangle Numbers
The $n$th term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are: $$1, 3, 6, 10, 15,...
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John Rhodes
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August 15, 2023
Pandigital Prime
We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once. For example, $2143$ is a...
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John Rhodes
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August 15, 2023
Champernowne’s Constant
An irrational decimal fraction is created by concatenating the positive integers: $$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$ It can be seen that the $12$th digit of the fractional part is $1$....
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John Rhodes
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August 15, 2023
Integer Right Triangles
If $p$ is the perimeter of a right angle triangle with integral length sides, $\{a, b, c\}$, there are exactly three solutions for $p = 120$. $\{20,48,52\}$,...