Euler problems/61 to 70
Problem 61
Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
Solution:
problem_61 = undefined
Problem 62
Find the smallest cube for which exactly five permutations of its digits are cube.
Solution:
problem_62 = undefined
Problem 63
How many n-digit positive integers exist which are also an nth power?
Solution: Since dn has at least n+1 digits for any d≥10, we need only consider 1 through 9. If dn has fewer than n digits, every higher power of d will also be too small since d < 10. We will also never have n+1 digits for our nth powers. All we have to do is check dn for each d in {1,...,9}, trying n=1,2,... and stopping when dn has fewer than n digits.
problem_63 = length . concatMap (takeWhile (\(n,p) -> n == nDigits p))
$ [powers d | d <- [1..9]]
where powers d = [(n, d^n) | n <- [1..]]
nDigits n = length (show n)
Problem 64
How many continued fractions for N ≤ 10000 have an odd period?
Solution:
problem_64 = undefined
Problem 65
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
Solution:
problem_65 = undefined
Problem 66
Investigate the Diophantine equation x2 − Dy2 = 1.
Solution:
problem_66 = undefined
Problem 67
Using an efficient algorithm find the maximal sum in the triangle?
Solution:
problem_67 = undefined
Problem 68
What is the maximum 16-digit string for a "magic" 5-gon ring?
Solution:
problem_68 = undefined
Problem 69
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
Solution:
problem_69 = undefined
Problem 70
Investigate values of n for which φ(n) is a permutation of n.
Solution:
problem_70 = undefined