Digit Fifth Powers
Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits: \begin{align} 1634 &= 1^4 + 6^4 + 3^4...
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John Rhodes
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August 15, 2023
Consecutive Prime Sum
The prime $41$, can be written as the sum of six consecutive primes: $$41 = 2 + 3 + 5 + 7 + 11 + 13.$$ This...
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John Rhodes
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August 15, 2023
Prime Permutations
The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are...
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John Rhodes
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August 15, 2023
Self Powers
The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$. Find the last ten digits of the series, $1^1 + 2^2 + 3^3 +...
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John Rhodes
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August 15, 2023
Distinct Primes Factors
The first two consecutive numbers to have two distinct prime factors are: \begin{align} 14 &= 2 \times 7\\ 15 &= 3 \times 5. \end{align} The first three...
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John Rhodes
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August 15, 2023
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...