615

Given that x and n are integers, find all possible solutions, i.e. pairs of \left (x,n\right ) such that the sum of the square of x and 615 is equal to two raised to the nth power or x^2+615=2^n.
Source: mathcontest.olemiss.edu 10/31/2011

SOLUTION
x^2+615=2^n implies that x^2 is odd which in turn implies that x is odd. Let’s take a look at a few values of x^2 where x is an odd integer:
\cdots
\left (-9\right )^2=81
\left (-7\right )^2=49
\left (-5\right )^2=25
\left (-3\right )^2=9
\left (-1\right )^2=1
1^2=1
3^2=9
5^2=25
7^2=49
9^2=81
\cdots
The pattern shows that x^2 ends in 1, 5, or 9. Hence, it follows that x^2+615 ends in 6, 0, or 4 respectively.

Let’s take a look at a few values of 2^n where n is an integer:
\cdots
2^{-2}=\frac{1}{2^2}
2^{-1}=\frac{1}{2}
2^0=1
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
\cdots
The non-positive integers n produce too small a value compared to x^2+615 so we will ignore them. Instead, we investigate the positive integers n which produce 2^n values that end in 2, 4, 8, or 6. For the equation x^2+615=2^n to hold true, 2^n must end in either 4 or 6. It follows from the above pattern that n must be even or n=2k for some positive integer k.

We write
x^2+615=2^{2k}
615=2^{2k}-x^2
=\left (2^k\right )^2-x^2
=\left (2^k+x\right )\left (2^k-x\right )

615=1\cdot 3\cdot 5\cdot 41
The possible products that produce 615 are:
1\cdot 615=615
3\cdot 205=615
5\cdot 123=615
41\cdot 15=615
Which one is the correct one? Let’s add the factors to figure out:
\left (2^k+x\right )+\left (2^k-x\right )=2\left (2^k\right )
We now have a guideline to pick the correct product: the sum of its factors is a power of 2.
The sums of the factors are:
1+615=616        Not a power of 2
3+205=208        No
5+123=128        Yes a power of 2
41+15=56           No

We have
128=2\left (2^k\right )
\frac{128}{2}=2^k
64=2^k
2^6=2^k
k=6

Value of \mathbf n
n=2k=2\cdot 6=12

Values of \mathbf x
x^2+615=2^{12}
x^2+615=4096
x^2=4096-615
x^2=3481
x=\pm \sqrt{3481}
x=\pm 59

Answer: \left (-59,12\right ) and \left (59,12\right ).

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.

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