## 嵇爾的吐槽

#没事画轮子的嵇尔不定期的(W)碎(E)碎(B)念(B)和(L)吐(O)槽(G)

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Problem 63 Powerful digit counts 2019-11-13 08:30:00

The 5-digit number, 16807=$7^5$, is also a fifth power. Similarly, the 9-digit number, 134217728=$8^9$, is a ninth power.

How many n-digit positive integers exist which are also an nth power?

Problem 62 Cubic permutations 2019-07-12 08:30:00

The cube, 41063625 ( $345^3$ ), can be permuted to produce two other cubes: 56623104 ( $384^3$ ) and 66430125 ( $405^3$ ). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

 Triangle $P_{3,n}=n(n+1)/2$ 1, 3, 6, 10, 15, … Square $P_{4,n}=n2$ 1, 4, 9, 16, 25, … Pentagonal $P_{5,n}=n(3n−1)/2$ 1, 5, 12, 22, 35, … Hexagonal $P_{6,n}=n(2n−1)$ 1, 6, 15, 28, 45, … Heptagonal $P_{7,n}=n(5n−3)/2$ 1, 7, 18, 34, 55, … Octagonal $P_{8,n}=n(3n−2)$ 1, 8, 21, 40, 65, …

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
2. Each polygonal type: triangle ($P_{3,127}=8128$), square ($P_{4,91}=8281$), and pentagonal ($P_{5,44}=2882$), is represented by a different number in the set.
3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

Problem 60 Prime pair sets 2019-06-27 08:30:00

The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.

Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime.



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# 赞服地铁 time ago 2019-03-25 17:34:00