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


  • Director at ElitesGrid | CAT 2016 - QA : 99.94, LR-DI - 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= 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

    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


Log in to reply
 

Looks like your connection to MBAtious was lost, please wait while we try to reconnect.