Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025  385 = 2640. Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.…

It is possible to show that the square root of two can be expressed as an infinite continued fraction.  2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213... By expanding this for the first four iterations, we get: 1 + 1/2 = 3/2 = 1.51 + 1/(2 + 1/2) = 7…

215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26. What is the sum of the digits of the number 21000? 题目大意: 题目大意: 215 = 32768 并且其各位之和为 is 3 + 2 + 7 + 6 + 8 = 26. 21000 的各位数之和是多少? // (Problem 16)Power digit sum // Completed on Sun, 17 No…

Work out the first ten digits of the sum of the following one-hundred 50-digit numbers. 371072875339021027987979982208375902465101357402504637693767749000971264812489697007805041701826053874324986199524741059474233309513058123726617309629919422133635…

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 + 21215 = 7 + 22221 = 3 + 23225 = 7 + 23227 = 19 + 22233 = 31 + 212 It turns out that the conjecture was false. What…

The nth term of the sequence of triangle numbers is given by, tn = 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…

The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145: 1! + 4! + 5! = 1 + 24 + 120 = 145 Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to 169; it tu…

The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3. Fi…

The decimal number, 585 = 10010010012(binary), is palindromic in both bases. Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2. (Please note that the palindromic number, in either base, may not include le…

The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime. There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97. How many circular primes are there…