Distinct Powers
Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$: \begin{matrix} 2^2=4, &2^3=8, &2^4=16, &2^5=32\\ 3^2=9, &3^3=27, &3^4=81, &3^5=243\\...
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John Rhodes
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August 15, 2023
Number Spiral Diagonals
Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows: 21 22 23 24...
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John Rhodes
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August 15, 2023
Quadratic Primes
Euler discovered the remarkable quadratic formula: $n^2 + n + 41$ It turns out that the formula will produce $40$ primes for the consecutive integer values $0...
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John Rhodes
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August 15, 2023
Reciprocal Cycles
A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given: \begin{align} 1/2 &= 0.5\\ 1/3...
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John Rhodes
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August 15, 2023
$1000$-digit Fibonacci Number
The Fibonacci sequence is defined by the recurrence relation: $F_n = F_{n – 1} + F_{n – 2}$, where $F_1 = 1$ and $F_2 = 1$. Hence...
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John Rhodes
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August 15, 2023
Lexicographic Permutations
A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the...
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John Rhodes
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August 15, 2023
Non-Abundant Sums
A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper...
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John Rhodes
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August 15, 2023
Names Scores
Using names.txt (right click and ‘Save Link/Target As…’), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out...