A number theory problem by Bala vidyadharan

Number Theory Level 3

Given that a , b , c , d a,b,c,d belongs to positive integers and a 5 = b 4 { a }^{ 5 }={ b }^{ 4 } and c 3 = d 2 { c }^{ 3 }={ d }^{ 2 } and c − a = 19 c-a=19 .Then find the numerical value of d − b d-b .


The answer is 757.

Bala Vidyadharan by Bala vidyadharan , 20, India

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1 solution

We are looking after number c as square number and number a as a quartic and their difference is 19. Minimum possible solution is c = 100 c=100 and a = 81 a=81 ( c − a = 19 c-a=19 ). From ( 1 0 2 ) 3 = ( 1 0 3 ) 2 (10^2)^3=(10^3)^2 follow d = 1000 d=1000 and from ( 3 4 ) 5 = ( 3 5 ) 4 (3^4)^5=(3^5)^4 follow b = 243 b=243 . So d − b = 757 d-b=757 .

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