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Difference of Two Expressions equal n (Posted on 2024-01-30)

Find all positive real numbers x and nonnegative integers n such that:

xⁿ - ⌊x⌋ = n
*** ⌊n⌋ is the floor of n, which is equal to
the greatest integer less than or equal to n.

No Solution Yet Submitted by K Sengupta

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answer

Comment 1 of 1

n = 0 works with 1 <= x < 2

n = 2 works with x = 2 or x = sqrt(3) = 3^(1/2)

n = 3 works with x = 2^(2/3) = 4^(1/3)

n = 4 works with x = 5^(1/4)

n = 5 works with x = 6^(1/5)

n = 6 works with x = 7^(1/6)

...

n = k works with x = (k+1)^(1/k)

Posted by Charlie on 2024-01-30 13:37:21

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