) Does there exist a sequence of 10^100 consecutive composite numbers?
Yes , there exist a sequence of composite numbers.
Let = 10^ 100
t is possible to have N consecutive composite numbers. The series -
(N+2)!+2,(N+2)!+3, (N+2)!+4,......., (N+2)!+(N+2)
It is easy to prove it because, N! is a multiplication of all numbers from 1 to N, so you can see
(N+2)!+2(N+2)!+2 will give you 2 as common factor
(N+2)!+3(N+2)!+3 will give you 3 as common factor
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(N+2)!+(N+2)(N+2)!+(N+2) will give you (N+2) as common factor
showing N consecutive composite numbers.
) Does there exist a sequence of 10^100 consecutive composite numbers?
(1) Does there exist a sequence of 10100 consecutive composite numbers? (2) Let Pn be the nth prime number. Show that pn < 22".
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
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