Problem It is possible to show that the square root of two can be expressed as an infinite continued fraction. By expanding this for the first four iterations, we get: 1 + 1/2 = 3/2 = 1.5 1 + 1/(2 + 1/2) = 7/5 = 1.4 1 + 1/(2 + 1/(2 + 1/2)) = 17/12 …
Problem Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed. It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that …
Problem Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction. If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, …
Problem Euler’s Totient function, ?(n) [sometimes called the phi function], is used to determine the number of numbers less than 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, ?(9)=6. It can be seen that n=6 produces …
Problem Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction. If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, …
Problem Take the number 192 and multiply it by each of 1, 2, and 3: 192 x 1 = 192 192 x 2 = 384 192 x 3 = 576 By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3) The same …
Problem 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, …
Problem Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. Solution Initially I started solving this problem by creating all permutations of 19 digits numbers which fits the required pattern and iteratively square rooting them to find a square root which is a whole number. Unfortunately …
Problem The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s. We shall consider fractions like, 30/50 = 3/5, to be trivial examples. There are exactly four non-trivial examples of this type of …
Problem The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 26972593-1; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2p-1, have been found which contain more digits. However, in 2004 there was found a massive non-Mersenne prime which …
By continuing to use the site, you agree to the use of cookies. more information
The cookie settings on this website are set to "allow cookies" to give you the best browsing experience possible. If you continue to use this website without changing your cookie settings or you click "Accept" below then you are consenting to this.