Find value of A, B and C such that (A^B)+(B^C)+(C^A) is equal to given integer X

Question

X is positive integer.I have to find 3 distinct numbers A, B and C such that 0 ≤ A,B,C < 2³⁰ and (A^B)+(B^C)+(C^A)=X where '^' is a bitwise XOR operator

Answer 1

Disclaimer: I'm not an mathematician.

There are several solutions, this finds one of them.

1. take the X and convert it to binary (e.g 20220408 - yesterday's date)

   X=1001101001000100111111000

2. If the last digit is 1 (X is odd), there is no solution

3. Discard the last 0 (divide by 2 or shift right)

   X=100110100100010011111100

4. Create 3 binary numbers: start with zeroes and put each 1 bit from X to either a or b or c. It does not matter which one. Just for fun let's do it in a round-robin matter:

   a=100000100000000010010000

   b=000100000100000001001000

   c=000010000000010000100100

   Note: if there is only one 1 bit (X is a power of 2), a,b,c won't be distinct, because 2 of them will be zero.

Check the result:

a=int('100000100000000010010000', 2)
b=int('000100000100000001001000', 2)
c=int('000010000000010000100100', 2)
print((a^b)+(b^c)+(c^a)) # 20220408