In a prior post the property of all prime numbers greater than 3 can be expressed as 6n +/-1. A question that may arise from this property is whether all multiples of 6 are adjacent to a prime. The short answer is no, but one needs to review all multiples of 6 up to 120 before one finds the first multiple of 6 that is not adjacent to a prime number.
Examples:
1 x 6 = 6 is adjacent to primes 5 and 7
2 x 6 = 12 is adjacent to primes 11 and 13
3 x 6 = 13 is adjacent to primes 17 and 19
4 x 6 = 24 is adjacent to prime 23
At 20 x 6 = 120 is adjacent to 119 (composite 7 x 17) and 121 (composite 11 x 11)
As with many patterns of integers, it is always worth checking the On-Line Encyclopedia of Integer Sequences. We find that sequence {120,144,186,204,216,246,288,300...}
multiples of 6 that are not a prime number +/- 1 is sequence A259826.
Note: the first entry at 120 occurs after the first prime number gap >8 which is between 113 and 127.
Related posts on prime numbers:
https://jamesmacmath.blogspot.com/2020/05/prime-number-gap-conjectures.html
https://jamesmacmath.blogspot.com/2020/05/prime-number-gaps.html
https://jamesmacmath.blogspot.com/2020/05/prime-numbers-property-rediscovered.html
https://jamesmacmath.blogspot.com/2020/05/twin-prime-sandwich.html
No comments:
Post a Comment