If a, b, c are integer powers of 2, prove that equation
a^3 +b^4 = c^5
Answers
Step-by-step explanation:
If a is a power of two, then a^3 is a power of two whose exponent is a multiple of three. Likewise, b^4 is a power of two whose exponent is a multiple of 4, and c^5 is a power of two whose exponent is a multiple of 5.
The only way for the sum of two powers of 2 to be another power of two is if the two initial powers are the same, for then 2^n + 2^n = 2^(n+1)
It follows that to find suitable a, b, and c, we want to find a multiple of three that is also a multiple of 4 (that is, a multiple of twelve) that is also 1 less than a multiple of 5.
The multiples of 12 that are 1 less than a multiple of five are 24, 84, 144, … (increasing by 60).
From 24 + 1 = 25, we find the solution a=2^8, b=2^6, c=2^5
From 84 + 1 = 85, we find the solution a=2^28, b=2^21, c=2^17
From 144+1=145, we find the solution a=2^48, b=2^36, c=2^29
In general, there are infinitely many solutions; the all have the form
a = 2^(20n-12), b=2^(15n-9), c=2^(12n-7) for n=an integer.