How many satisfy?

Number Theory Level pending

How many natural numbers, $n$ , satisfy all the criteria below?

$2^n - 1$ is a prime number
The $n$ th triangle number is an even number
It has exactly 18 factors
The sum of the digits is a multiple of 7
It has at least 4 digits
It is a square number
The difference between $n^5$ and $n^4$ is a square number itself
It is a term in the Fibonacci sequence

The answer is 0.

1 solution

Stephen Mellor
Oct 8, 2017

Looking at the third and sixth criteria:

As they have 18 factors, the numbers are composite → All composite numbers have factors which are split up into pairs → If a number is a square number, one factor is repeated, hence only showing as one factor for that pair → A square number therefore has an odd number of factors → The numbers asked for are square and have an even number of factors

Since, these two criteria cannot both be satisfied simultaneously, the answer is $\boxed{0}$

How many satisfy?

The answer is 0.

1 solution

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