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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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= factors1= 91
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
PS Solve These Questions and Comment your answers
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