# Number of ways in which a natural number can be expressed as Sum of two perfect squares - Hemant Malhotra

• CASE 1 - when N is not a perfect square and prime factors in 4k+1

Example - x^2 + y^2= 5^2 × 13^3

Here 5 and 13 are both are both of form 4k+1 so integral solutions possible and  number of positive integral solutions

= factors of 5^2 ×13^3 = 12 and total integral solutions =4×12=48

Case 2- when N is of form perfect square

x^2+y^2= 5^2 ×13^2

here number is perfect square and 5 and 13 both are in 4n+1 form so

positive integral solutions= factors-1= 9-1

and  total integral solutions= 4 × 8 + 4 = 36 we will consider only 4k+1 form of prime factors

Extra addition  of 4 is for x=+-5*13 and y=+-5*13

Case 3- when x^2+ y^2= 3^2 × 7^3

here 3 and 7 are 4 k+3 form and no 4 k+1  form  so number of integral solutions = zero

Case 4 -   x^2 + y^2= 5^2 × 3^4

here 3 is 4 k+3 form and 5 is 4 k+1 form so

now 2 cases arise

Case 1 ) when 4k+3 form has odd power then number of integral solutions =0

Case 2)  when 4k+3 form has even power then ignore that and find number of factors of 4k+1 form

Example x^2 + y^2= 5^3 ×7^2

here 7 is 4 k+3 form but power is even so ignore that now find factors of 5^3 which is 4 so number of positive integral solutions =4

and total =4 ×4=16

CASE 5- when number is of perfect square form and no 4k+1 form also then number of integral solutions will be 4 only

Example=  x^2 + y^2 = 81 = 3^4

here number of integral solutions is 4

1) x^2 + y^2 = 80

2)  x^2 + y^2 = 80

3) x^2 + y^2 = 121

4) x^2 + y^2 = 126

5) x^2 + y^2 = 120

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