이경수 선생님의 수학실험실

Problem 51(Prime digit replacements) By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime. By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663..

Problem 50(Consecutive prime sum) 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 o..

Problem 49(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 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 increasin..

Problem 48(Self powers) The series, \(1^{1}+ 2^{2}+ 3^{3}+ ... + 10^{10}= 10405071317\). Find the last ten digits of the series, \(1^{1}+ 2^{2}+ 3^{3}+ ... + 1000^{1000}\). In Python: import time def selfpower(n): product = 1 for i in range(1, n + 1): product = (product * n) % (10 ** 10) return product result = 0 for i in range(1, 1000): result += selfpower(i) result = result % (10 ** 10) print(..

Problem 47(Distinct primes factors) The first two consecutive numbers to have two distinct prime factors are: \(14 = 2 \times 7\) \(15 = 3 \times 5\) The first three consecutive numbers to have three distinct prime factors are: \(644 = 2^{2} \times 7 \times 23 \) \(645 = 3 \times 5 \times 43 \) \(646 = 2 \times 17 \times 19 \). Find the first four consecutive integers to have four distinct prime..

Problem 46(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. \(9 = 7 + 2\times1^{2}\) \(15 = 7 + 2 \times 2^{2}\) \(21 = 3 + 2 \times 3^{2}\) \(25 = 7 + 2 \times 3^{2}\) \(27 = 19 + 2 \times 2^{2}\) \(33 = 31 + 2 \times 1^{2}\) It turns out that the conjecture was false. What is the smallest..

Problem 45(Triangular, pentagonal, and hexagonal) Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: Triangle \(T_{n}=\frac{n(n+1)}{2}\) 1, 3, 6, 10, 15, ... Pentagonal \(P_{n}=\frac{n(3n-1)}{2}\) 1, 5, 12, 22, 35, ... Hexagonal \(H_{n}=n(2n-1)\) 1, 6, 15, 28, 45, ... It can be verified that \(T_{285} = P_{165} = H_{143} = 40755\). Find the next triangle number ..

Problem 44(Pentagon numbers) Pentagonal numbers are generated by the formula, \(P_{n}=\frac{n(3n−1)}{2}\). The first ten pentagonal numbers are: \(1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...\) It can be seen that \(P_{4} + P_{7} = 22 + 70 = 92 = P_{8}\). However, their difference, \(70 − 22 = 48\), is not pentagonal. Find the pair of pentagonal numbers, \(P_{j}\) and \(P_{k}\), for which their s..