Answer to Question #289693 in Algebra for arnob

Question

Find the smallest positive integer a

 for which there exist distinct positive integers b,c,d

 such that a=b^2=c^3=d^4

.

Answer 1

a=b²=c³=d⁴;

maximum degree = $3*4=12$, so

x¹²=(x⁶)²=(x⁴)³=(x³)⁴, so

if x=1, then a=b=c=d=1, no distinct positive integers

if x=2, then 2¹²=(2⁶)²=(2⁴)³=(2³)⁴, so

a=4096, b=64, c=16, d=8

Answer: 4096