Задачи проекта Эйлера

41-50

Список всех задач

41. We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.

    What is the largest n-digit pandigital prime that exists?

42. The n^th term of the sequence of triangle numbers is given by, t_n = ½n(n+1); so the first ten triangle numbers are:

    1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

    By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t₁₀. If the word value is a triangle number then we shall call the word a triangle word.

    Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words?

43. The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.

    Let d₁ be the 1^st digit, d₂ be the 2^nd digit, and so on. In this way, we note the following:

    • d₂d₃d₄=406 is divisible by 2
    • d₃d₄d₅=063 is divisible by 3
    • d₄d₅d₆=635 is divisible by 5
    • d₅d₆d₇=357 is divisible by 7
    • d₆d₇d₈=572 is divisible by 11
    • d₇d₈d₉=728 is divisible by 13
    • d₈d₉d₁₀=289 is divisible by 17

    Find the sum of all 0 to 9 pandigital numbers with this property.

44. 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, ...

    It can be seen that P₄ + P₇ = 22 + 70 = 92 = P₈. However, their difference, 70 - 22 = 48, is not pentagonal.

    Find the pair of pentagonal numbers, P_j and P_k, for which their sum and difference are pentagonal and D = |P_k - P_j| is minimised; what is the value of D?

45. Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

    Triangle		Tn=n(n+1)/2		1, 3, 6, 10, 15, ...
    Pentagonal		Pn=n(3n-1)/2		1, 5, 12, 22, 35, ...
    Hexagonal		Hn=n(2n-1)		1, 6, 15, 28, 45, ...

    It can be verified that T₂₈₅ = P₁₆₅ = H₁₄₃ = 40755.

    Find the next triangle number that is also pentagonal and hexagonal.

46. It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

    9 = 7 + 2$\cdot$1²
    15 = 7 + 2$\cdot$2²
    21 = 3 + 2$\cdot$3²
    25 = 7 + 2$\cdot$3²
    27 = 19 + 2$\cdot$2²
    33 = 31 + 2$\cdot$1²

    It turns out that the conjecture was false.

    What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

47. The first two consecutive numbers to have two distinct prime factors are:

    14 = 2 $\cdot$ 7
    15 = 3 $\cdot$ 5

    The first three consecutive numbers to have three distinct prime factors are:

    644 = 2² $\cdot$ 7 $\cdot$ 23
    645 = 3 $\cdot$ 5 $\cdot$ 43
    646 = 2 $\cdot$ 17 $\cdot$ 19.

    Find the first four consecutive integers to have four distinct prime factors. What is the first of these numbers?

48. The series, 1¹ + 2² + 3³ + ... + 10¹⁰ = 10405071317.

    Find the last ten digits of the series, 1¹ + 2² + 3³ + ... + 1000¹⁰⁰⁰.

49. 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 prime, and, (ii) each of the 4-digit numbers are permutations of one another.

    There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

    What 12-digit number do you form by concatenating the three terms in this sequence?

50. The prime 41, can be written as the sum of six consecutive primes:

    41 = 2 + 3 + 5 + 7 + 11 + 13

    This is the longest sum of consecutive primes that adds to a prime below one-hundred.

    The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953.

    Which prime, below one-million, can be written as the sum of the most consecutive primes?