MEI Maths Item of the Month
November 2013
A prime example
Given the list of the first n primes, 2,3,5,..., the product of the primes plus one will be coprime
to the list of primes used.
e.g.
E1 = 2 + 1
=3
and 3 is coprime to 2.
E2 = 2×3 + 1
=7
and 7 is coprime to 2 and 3.
Will En always be a prime number?
Will En ever be a square number?
Solution
The first value of En that is not a prime number is E6.
E6 = 2×3×5×7×11×13 + 1
= 30031
= 59×509
En will never be a square number.
En = p1× p2×…× pn + 1
If En is a square number this can be written as:
m² = p1× p2×…× pn + 1
p1× p2×…× pn = m² – 1
= (m + 1)(m – 1).
(m + 1) (m – 1) must either be both odd or both even for this result to hold.
p1× p2×…× pn has a single even factor of 2. (m + 1) and (m – 1) cannot both be odd as this
would result in p1× p2×…× pn being odd. (m + 1) and (m – 1) cannot both be even as this
would mean 4 was a factor of p1× p2×…× pn.
Therefore En will never be a square number.
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