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kaliprasad
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Find all pairs of integers n and k such that $n!+8 = 2^k$
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kaliprasad said:Find integers n and k such that $n!+8 = 2^k$
I really don't like that ISPOILER effect! :sick:Klaas van Aarsen said:From inspection I can find $n=4,\,k=5$.
As a bonus I can see that $n=5,\,k=7$ works as well.
I kind of suspect that you intended to find all such integers, did you? Ah well, that was not the question.
It's new in Xenforo. I felt I just had to try it out.Opalg said:I really don't like that ISPOILER effect! :sick:
Klaas van Aarsen said:From inspection I can find $n=4,\,k=5$.
As a bonus I can see that $n=5,\,k=7$ works as well.
I kind of suspect that you intended to find all such integers, did you? Ah well, that was not the question.
If you click on the blurred sections in the above comments, they will become clear and reveal the solutions.Arlynnnn said:Where's the solutions?
In this equation, "n" represents an unknown number and "k" represents the exponent of the power of 2.
To solve for the values of n and k, you can use algebraic manipulation and substitution. First, subtract 8 from both sides of the equation to get n=2^k-8. Then, you can substitute this value of n into the second equation, n!+8=2^k, to get (2^k-8)!+8=2^k. From here, you can use mathematical techniques such as factoring, expanding, and simplifying to solve for the values of n and k.
Yes, there can be multiple solutions for n and k in this equation. This is because there are many different combinations of numbers that can be raised to a power of 2 and then have 8 added to them to equal another power of 2. For example, if n=4 and k=4, the equation becomes 4+8=2^4 and 4!+8=2^4, but if n=3 and k=5, the equation also holds true as 3+8=2^5 and 3!+8=2^5.
There is not a specific method or formula for solving this equation. It requires a combination of algebraic manipulation and mathematical techniques such as factoring and simplifying to solve for the values of n and k.
This equation can be used in various fields of science and mathematics, such as computer science, cryptography, and number theory. It can also be used to solve problems involving combinations and permutations, as well as in the study of prime numbers and their properties.