You could have directly assumed the odd / even logic right at the beginning. Since the answer is odd, 8^(y-8) must be 8^(0). Thus directly y = 8. Then apply the remaining logic to derive the value of x.

Excellent , again well presented. Thank you.

OBS. 8^(a+y-8)-8^(y-8)= 4095; 8^(y-8)x(8^ a-1)=4095. Etc.

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Thank you,

Fairly simple to do in your head . Most people know 2^10 =1024 so,2^12 =4096 which is 4095 +1 2^12 =2^3^4=8^4 and 1= anything ^0 including 8^0 => 8^4-8^0 =4095 and the rest is easy The driving factor wa"what is the lowesrpower of 2 which is greater than 4095 but divisible by 3 ?' Divisible by three means it is an power of 8,

Greetings. Great job. Absolutely brilliant.

Μπράβο σου. Είσαι καταπληκτική. Από Ελλάδα. Greece

Elegant

Good job. Thank you

The solution set is x, y = {12, 8} Working Method. My approach would be to rewrite the right side with a base of 8 - since 4095 = 4096 - 1. 4096 = 2^12 or 8^4; and 1 = 8^0. Therefore; 4095 = 8^4 - 8^0 Accordingly; The entire expression can be presented as: 8^(x - 8) - 8^(y - 8) = 8^4 - 8^0 From this, we can equate exponents and solve the resulting linear equations, thus: x - 8 = 4, and y - 8 = 0 If x - 8 = 4, x = 4 + 8; x = 12 If y - 8 = 0, y = 8 + 0; y = 8 Therefore, the final solution set is x, y = {12, 8} Checking: x = 12, and y = 8 8^(12 - 8) - 8^(8 - 8) = 4095 8^4 - 8^0 = 4095 4096 - 1 = 4095 (Checked).

A unique solution in my style... , let u=8^((x-8)/2) , v=8^((y-8)/2) , u^2=8^(x-8) , v^2=8^(y-8) , u^2-v^2=4095 , (u+v)(u-v)=4095 , etc. , 4095=105*39 , let u+v=105 , u-v=39 , u+v+u-v=105+39 , 105+39=144 , -> 2u=144 , u=72 , u+v=105 , v=105-72 , v=33, u^2=72^2 , u^2=5184 , v^2=33^2 , v^2=1089 , for x , -> 8^(x-8)=5184 , (x-8)*ln8=ln5184 , x-8=ln5184/ln8 , x-8=4.11328 , x=4.11328+8 , x=12.11328 , for y , -> 8^(y-8)=1089 , (y-8)=ln1089/ln8 , y-8=3.36293 , y=11.36293 , test , 8^(x-8)-8^(y-8)=8^(12.11328-8)-8^(11.3693-8) , 8^(12.11328-8)-8^(11.3693-8)=8^(4.11328) - 8^(3.36293) , test -> , 8^4.11328=5184 , 8^3.36293=1089 , 5184-1089=4095 , same , OK , solu. , x=12.11328 , y=11.36293 , /// Large numbers can be tamed by using logarithms... /// , By dividing the problem into products of 4095, there are many, many solutions... etc. , 4095=5*3*7*3*13 , -> 15*273=4095,,, ,

Thank you