unanswered_questions.txt

Problem 60: Prime pair sets
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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problem 61 Cyclical figurate numbers

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.

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).
Each polygonal type: triangle (P(3,127)=8128), square (P(4,91)=8281), and pentagonal (P5,44=2882), is represented by a different number in the set.
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.


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Problem 70 Totient permutation

Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.

The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.

Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.

Find the value of n, 1 < n < 10^7, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.