sequence analysis of prime number related sequence

Prime numbers fascinate mathematicians, and Jamie clearly isn't one.

The Legendre conjecture

has yet to be proven correct (though it does seem likely to be correct).

I once met a mathematician who got his Ph.D. for proving a weaker conjecture, which still took an impressive armoury of advanced math.

--
Bill Sloman, Sydney

As far as expertise, knowledge ad skill goes, he seems far ahead of everyone else.

--
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Hi,

To prove weaker conjectures directly about prime numbers requires more mathematical tools, but to figure out more general patterns in prime numbers, which could implicity prove the weaker conjectures probably only requires statistics.

cheers, Jamie

Hi,

Here is a way to use the sequences above to calculate the nth composites:

Sequence 1 corresponds to composites with least prime factor 7. Sequence 2 corresponds to composites with least prime factor of 11.

(there is a sequence for each least prime factor)

reorganized sequence2 (just reversed the sequence and shifted by 1):

53,34,31,12,9,22,3,16,13,-6,-9,4,17,-2,11,8,-11,2,-1,12,41,38,19,16,-3,-6,23,36,33,46,27,24,37,18,31,44,41,22,19,32,13,26,23,4,1,-18,27,8

Take all values in the sequence and multiply by 11/8=1.375:

gives the sequence:

72.875,46.75,42.625,16.5,12.375,30.25,4.125,22,17.875,-8.25,

-12.375,5.5,23.375,-2.75,15.125,11,-15.125,2.75,

-1.375,16.5,56.375,52.25,26.125,22,-4.125,

-8.25,31.625,49.5,45.375,63.25,37.125,33,50.875,24.75,42.625,

60.5,56.375,30.25,26.125,44,17.875,35.75,31.625,5.5,1.375,

-24.75,37.125,11

find the value x of the nth composite with least prime factor 11:

ie for n = 46, x = 2189.

1. n % 48 = 46 % 48 = 46
2. The 46th value z in sequence 2 = -24.75
3. ((8/385)*x)-((8/385)*-24.75)=46. x=2189.

The values 8/385 and 11/8 are given by:

ie: 11/8 = (prime(5)=11) / (numerator of 8/385 below)

8/385 = ((7-1)*(5-1)*(3-1)*(2-1))/(2*3*5*7*11)

Patterns in the sequence z are related to distribution of the primes.

cheers, Jamie

The distribution of the primes is driven by the fact that all the non-prime numbers in between the primes sit on regular patterns - of the smaller prime numbers that are multiplied together to get them.

Looking for regularities and patterns in the distribution of the primes themselves is a demonstration that you don't actually appreciate what a prime number is.

--
Bill Sloman, Sydney

snipped-for-privacy@ieee.org wrote in news: snipped-for-privacy@googlegroups.com:

Decidedly making any of those results NON-prime. By definition.

I think you missed the point.

I have restored the first part of the sentence which you had snipped - presumably because you hadn't got the point.

The primes sit in holes within the regular patterns formed by non-prime numbersprecisely because they are prime numbers, and can't thus themselves be a part of any kind of regular pattern.

--
Bill Sloman, Sydney

Hi,

Here are some basic example patterns for sequence 2 in the first post of the thread:

sequence2: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37,

24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11,

-2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

That sequence of 48 values is symmetrical, ie For values x and y at positions a and b:

x + y = 35, where a = 48 - (b-1)

Where 35 is given by the denominator of ((7-1)*(5-1)*(3-1)*(2-1))/(2*3*5*7) = 8/35.

Another pattern in the sequence:

Take every third value in the sequence, giving three new sequences:

27,4,13,22,31,24,33,-6,19,12,-11,-2,-9,16,9,34.

-18,23,32,41,18,27,36,-3,38,-1,8,17,-6,3,12,53.

1,26,19,44,37,46,23,16,41,2,11,4,13,22,31,8.

The sum of each of those sequences is:

216,280,344.

And 280-216=344-280=64

There are lots more patterns too, also patterns between the sequences I think.

If you find enough patterns to construct the sequences I posted, then they can be used to give the numbers with least prime factor corresponding to the sequence, ie sequence 2 corresponding to the sequence of values with least prime factor 11:

121 143 187 209 253 319 341 407 451 473 517, ...

and also the prime number distribution is encoded in that up to a range, since ie for numbers with least prime factor 11, divided by 11 gives the prime numbers in the range 11 to 11^2.

cheers, Jamie