Consecutive $n$ composite numbers

How can we generalize any theorem for finding $n$ consecutive positive composite numbers?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

For $n \ge 1$ consider the $n$ -term sequence $(n + 1)! + 2, (n + 1)! + 3, (n + 1)! + 4, ..... , (n + 1)! + (n + 1).$

Then the $k$ th term is divisible by $k + 1$ for $1 \le k \le n,$ and thus all $n$ terms are composite.

Brian Charlesworth - 5 years, 7 months ago

Are these terms the lowest possible numbers as well? I guess no, so can we generalize them for lowest possible terms?

Abhijeet Verma - 5 years, 7 months ago

I'm not sure if there will be a formula for the lowest possible terms for a given $n.$ As you point out $8,9,10$ are the lowest for $n = 3.$ For $n = 2$ it would be $8,9,$ (which matches the sequence I mentioned above). For $n = 4$ it would be $24,25,26,27$ and for $n = 5$ it would be $24,25,26,27,28.$ For $n = 6$ it would be $90,91,92,93,94,95$ and for $n = 7$ it would be $90,91,92,93,94,95,96.$

Since prime gaps for primes in excess os $2$ are multiples of $2$ we will find that the lowest sequences for $n = 2k$ and $n = 2k + 1$ will have the same lowest term. Beyond that observation, I think it would be difficult to establish any general formula since the number of consecutive composites depends on the sizes of the prime gaps, (i.e., difference between successive primes), which don't follow any identifiable pattern.

Brian Charlesworth - 5 years, 7 months ago

As if I have to find three consecutive positive composite numbers, the lowest are 8,9 and 10.

Abhijeet Verma - 5 years, 7 months ago

@Brian Charlesworth Sir, please help again!

Abhijeet Verma - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Consecutive nnn composite numbers

Comments

Consecutive $n$ composite numbers