The answer is correct but the method is wrong! Many youtubers also make the same mistake. You will realize your mistake when applying your method to the problem x^u - y^t = 17. You will never find the following solution of the problem 5^2-2^3=17. You also cannot find the a simple solution of the above equation is 18^1-1^18=17.The correct way to do it is as follows: We realize that x and y cannot both be even numbers or odd numbers. - If x is even number and y is odd number, we find the a simple solution of the above equation is 18^1-1^18=17 with x=18, y=1; In case x,y >= 2 => x^y≡0 (mod 4) and y^x≡17≡1 (mod 4) => y^x+17≡2(mod 4)=>x^y - y^x ≠ 17. - So the following case remains, x is odd number and y is even number => we can easily find the fist solution x=3, y=4. We will prove this is the only solution. * First, we will prove that x must be smaller than y. By examining the function f(t)=ln(t)/t, we see that this function is inverse when t>=3 => If x^y = y^x+17 > y^x then xx>3 => x= 3+a and y = 4+b with b >= a >= 1 => P = x^y - y^x = (3+a)^(4+b) - (4+b)^(3+a) = (3+a).(3+a)^(3+b) - [1+(3+b)]^(3+a). Because 3+a =< 3+b => (3+a)^(3+b) >= (3+b)^(3+a) => P >= (3+a)(3+b)^(3+a) - [1+(3+b)]^(3+a) = [(3+b)^(3+a)].{(3+a) - [(1+3+b)/(3+b)]^(3+a)} = [(3+b)^(3+a)].{(3+a) - [(1+1/(3+b)]^(3+a)}. And because 3+a 1 => [(1+ 1/(3+b)]^(3+a) < [(1+ 1/(3+b)]^(3+b) < e (e=2,71828….). => P > [(3+b)^(3+a)].{(3+a) - e}. And (3+b)^(3+a) >= 4^4 = 256 => P > 256.(4-e) > 17. Q.E.D
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