Vipul: Created page with '==Statement== For every positive integer $m$ , there exists a sequence of $m$ consecutive integers all of which are positive. Thus, there exists a [[prime g…'

2010-02-09T02:45:19Z

Created page with '==Statement== For every positive integer <math>m</math>, there exists a sequence of <math>m</math> consecutive integers all of which are positive. Thus, there exists a [[prime g…'

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==Statement==

For every positive integer <math>m</math>, there exists a sequence of <math>m</math> consecutive integers all of which are positive. Thus, there exists a [[prime gap]] between consecutive primes that is greater than <math>m</math>.

==Proof==

Let <math>n = m + 1</math>. Consider the integers <math>n! + 2, n! + 3, \dots, n! + n</math>. For each <math>2 \le i \le n</math>, <math>n! + i</math> is divisible by and strictly larger than <math>i</math>, hence is composite. Further, the sequence has length <math>n - 1 = m</math>.

There exist arbitrarily large prime gaps - Revision history

Vipul: Created page with '==Statement== For every positive integer m, there exists a sequence of m consecutive integers all of which are positive. Thus, there exists a [[prime g…'

Vipul: Created page with '==Statement== For every positive integer $m$ , there exists a sequence of $m$ consecutive integers all of which are positive. Thus, there exists a [[prime g…'