Which numbers can be expressed as the sum of two squares?

All prime numbers which are sums of two squares, except 2, form this series: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, etc. Not only are these contained in the form 4n + 1, but also, however far the series is continued, we find that every prime number of the form 4n+1 occurs.

Can every number be written as the sum of two squares?

A composite number for which every factor can be written as a sum of squares can also be written as a sum of squares. Now, we know that any number that is composed of prime factors only of the form 4k+1 can be written as the sum of squares.

Which numbers can be expressed as sum of two cubes in two different ways?

1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of : 10 and 9, cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729.

Can a number be represented as sum of two cubes?

m = (a3 + b3) = (c3 + d3) for distinct a , b , c , d . For example, If n = 25000 , m can be any of 1729 , 4104 , 13832 , or 20683 as these numbers can be represented as the sum of two cubes for two different pairs.

How do you find if a number is the sum of two squares?

A number can be represented as a sum of two squares precisely when N is of the form n2∏pi where each pi is a prime congruent to 1 mod 4. If the equation a2+1≡a(modp) is solvable for some a, then p can be represented as a sum of two squares.

Which numbers can be expressed as the sum of three squares?

Sum of squares theorems are theorems in additive number theory concerning the expression of integers as sums of squares of other integers. For example, 30 = 1 2 + 2 2 + 5 2 30 = 1^2 + 2^2 + 5^2 30=12+22+52, so 30 can be expressed as a sum of three squares.

What is so special about 1729?

1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9 – cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729.

Is 13832 a hardy Ramanujan number?

Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.

What is the meaning of 1729?

How do you find the sum of two squares?

“Any positive number n is expressible as a sum of two squares if and only if the prime factorization of n, every prime of the form (4k + 3) occurs an even number of times.” By making use of the above theorem, we can directly find out if the given number n can be expressed as a sum of two squares.

How do you find the sum of two squares with NNP?

If a number pis a sum of two squares, then the numbers 4p;9p;16pand, in general, nnpwill be sums of two squares. Certainly, because p= aa+bb, we will have 4p= 4aa+4bb;9p= 9aa+9bb;16p= 16aa+16bband nnpp= nnaa+nnbb, which are similarly sums of two squares.

How to check if a number is a perfect square?

Binary Search Approach : Another method to check if is a perfect square, is by making use of Binary Search. The method remains same as that of a typical Binary Search to find a number.