Happy new year to all you Numberphiles …

Hi all. It’s been so busy with work that I haven’t had time to play about with spreadsheets and numbers for a wee while now.  I’m going to re-visit the old stuff and see what else I can turn up.

 

Have Fun with sums !!

Numberphile pages.

Check out the numberphile You tube videos if you like numbers. There’s an interesting derivation of Pi (as in 3.141 …) using a bowling ball and a snooker ball which intrigued me.

There’s also lots of videos on some of the more simple proofs in number theory such as why there are an infinite number of primes. They also have nice videos on the latest proof that there are an infinite number of primes separated by 70,000,000 (although they don’t actually go through the proof) and suggest that they may be able to get that down to 16 which is coming ever closer to proving the twin prime conjecture .. i.e. that there are an infinite number of twin primes ..

All and all it’s fun stuff to watch compiled by a group of college lecturers who are obviously passionate about their subject. So if Christmas TV listings don’t appeal then google ‘numberphile’ and enjoy!!

Happy Christmas. 2013

Maybe there are lots of different sequences that make up the primes?

Yesterday I was saying how x^2+x+k  needs a twin prime to start the ball rolling and if you read the blog you’ll remember I said that it didn’t workwell for (29,31) or (59,61) twins.

One that does work is 2x^2 + k.

When k=29 then when x= 1 2x^2 = 2*1. so 2×2+29 = 31

when x=2 you get 2x^2 = 8 so 2x^2+29 = 37.

The first few terms of this sequence = 29,31,37,47,61.

Type them into OEIS and you’ll find the sequence.

Maybe the prime sequence isn’t one sequence but a myriad of different sequences. Perhaps the job of a prime nut is to identify where any particular prime belongs.

Reflections on x^2+x+k as a prime generating polynomial.

let’s look at the x^2+x bit first. 

Take the example of when

x=1 –>  x^2+x = 2

when

x = 2 –> x^2+x = 6.

when

x= 3 –> x^2+x = 12 ….

if k = 41 then x^2+x+k between x= 0 and x= 3 gives 41,43,47,53.

the gap grows by 2 each time.

To get the first 2 prime results (41,43) you have to start at the smaller value of a twin prime.

The twin primes in this case are 41,43.

This is the already mentioned famous Euler polynomial.

So what if you start at  k = 43 and work the other way?  i.e. k- x^2+x.

You can imagine an axis of symmetry at 42

Now the sequence grows back towards the negative numbers.

43, 41, 37, 31, 23, 13, 1, -13, -29, -47, -67, -89,-113,-139 ………

Okay so by convention 1 isn’t a prime and there are lots of clever mathematicians who can offer robust reasons why it shouldn’t be. Personally I think it is a prime, if for no other reason than when you look at sequences such as this it saves the day more often than not.

when k = 17 you also get the same effect

17,19,23,29,37,47,59,73 …… when x^2+x+17 ….

and

19,17,13,7,-1,-11,-23,-37,-53,-71,-91 :( … when 19-x^2+x) .. I said in an earlier blog that I hate 91 .. here’s just one reason why ;)

 

I call this the reflection conjecture … just because it has a ring to it … :))

What’s worthy of note is that when x^2+x+29 is used  (twin primes now being 29,31) … the sequence runs aground by the second term .. 29,31,35 …

and when you do the 31-x^2+x you get no further

31,29,25 ..

so all twim primes of this type … the next being 59,61   are not good candidates for k values.

the (71,73) twin pair do better as 73-x^2+x than x^2+x+71 …

73,71,67,61,53,43,31,17,1,-17,-37,-59,-83 ….. 

What I’m getting at in this blog .. is that when seeking sequences work the prime numbers from right to left as well as left to right. You’ll be pleasantly surprised at how often the sequences line up.

BP 05/08/2012

Prime Directive

After a lifetime spent looking at primes it’s odd (no pun intended) how your life becomes governed by them. Today, for example, when I go to the gym I will stop when the calorie counter reaches a prime number. At this moment in time I find I can comfortably row for 10 minutes and hit 113 calories. The next prime number up from 113 is 127. This is the largest gap in the primes so far when starting from 2.

Here’s a thought. A Highly Composite Number (HCN) sits in the middle of most big gaps in the primes. In the case above 120 (2^3*3*5) sits in between 113 and 127.  Some primorial or partial primorial do sometimes allow a prime to snuggle right up close (211 – (a prime) is one off from 210 which is 2*3*5*7) but those primorials that don’t have a prime buddy create big holes in the prime sequence. In the primorial conjecture I call this the primorial influence.

I once had a choice of buying a house at 27 or 29. No contest,  it had to be 29. I grew up in No 23. My mother still lives there and she’s coming up 89.

Post script: Today at the gym. While sitting on a recumant bike and watching the ladies Olympic triatherlon road cycling stage. I managed 149 calories in 10 minutes. It nearly killed me. A top tip is not to try and keep up the same cadence as an Olympic cyclist.

My epitaph would have read he died while still in his prime.

The On line encyclopedia of integer sequences … ya what?

As a species what sets us apart from the from our nearest neighbours who still prefer to swing from the branches is the ability to categorize and classify everything we see around us.  Trying to stake a claim on posterity is a very human thing to do. We love to leave a mark. Something that will remain when we have gone back to the dust.  We may have at sometime in our lives been impressed by a shelf full of the many volumes that comprise the Encyclopedia Britannica or the chunky tombs like Wisden Cricketing almanac which have all the stats and scores of those great games in the past. For those wishing to record what we have achieved in the world of numeric sequences we have to rely on the OEIS … the online encyclopedia of integer sequences.  This ‘encyclopedia’ was something started by a guy called Neil Sloane almost 40 years ago. Well not the on-line bit but the integer sequences bit. Like all good ideas at the time it is something that’s outgrown its humble roots and taken on a life of its own.

Here’s how it works.

You dream up a sequence  of integers that are related in some way .. which may only be tenuous … or it may be some thing that springs from a well defined formula .. like y=x^2 .. and you create a list of results and you can put the results in the OEIS … then someone else comes along and if they think they’d like to put the sequence in too they can type the first few digits in to see if someone else has already thought of it … Having just re-read that last paragraph .. it all sounds a tad dull. But if you’re a prime nut then it’s the dog’s whatsits.
Sequences in the OEIS start with the letter A .. for example A000040 is a list of prime numbers. This page dedicated to this sequence links to loads of other sequences and gives all the learned papers on prime numbers as reference … a veritable vade mecum  of all things integer.

So the reason for mentioning this is because on Saturday  .. I entered a sequence that got ‘accepted’ by the editors of the site. They tweaked about with it but in essence they let it be. It came about by playing with Excel and generating several thousand results for a very famous prime generating polynomial.  x^2  + x + 41. This is one of the most prolific prime generators. The first 40 values of X for example are all prime!! Okay those who have little interest in primes may be non-plused but believe me .. that’s a corker of a formula. Anything that can tip-toe through the integers in a well defined way and reach 1681 before it comes across its first composite number is doing very very well.

Here’s what I noticed. When you get to 41^2 where the  prime sequence starts to fall apart and your lovely stream of contiguous primes comes to an end .. something very interesting happens to the semiprimes.   A semi-prime is just two primes multiplied together.  Euler who first published this polynomial was more than aware it couldn’t continue to generate primes past 41.  What I did was to embrace this fact and start to look at how the semi-primes where made up.  I started this work about 3 months ago now as it takes quite a while factorizing numbers to see if they’re prime or not. If numbers get very large then I would suggest you go to Alpertron’s ECM site as that will take the grunt out of doing it.

I noticed that the factors themselves started sequences that appeared regularly through the results …

Let me give you an example so as to explain myself a little better.

37	1447	1447
38	1523	1523
39	1601	1601
40	1681	41	41
41	1763	41	43
42	1847	1847
43	1933	1933
44	2021	43	47
45	2111	2111
46	2203	2203
47	2297	2297
48	2393	2393
49	2491	47	53
50	2591	2591
51	2693	2693
52	2797	2797
53	2903	2903
54	3011	3011
55	3121	3121
56	3233	53	61
57	3347	3347
58	3463	3463
59	3581	3581
60	3701	3701
61	3823	3823
62	3947	3947
63	4073	4073
64	4201	4201
65	4331	61	71

The leftmost column is the value of X in the equation x^2+x+41. So, for example,  if you take the value 60 you get (60*60) + 60 + 41 = 3701 .. which just happens to be prime.

But now look at the result when X = 41 .. here the result is 1763 .. which isn’t a prime number but is a number with two factors i.e. 41 and 43. This is because 41 * 43  = 1763.

Look down columns 3 and 4 and you get all the factors (41*41), (41*43),(43*47),(47*53)(53*61)….. ( I can feel a ‘so what’ moment coming on .. :) )

Well these factors are the start of the sequence for x^2+x+41  when x = 1 . (See below)

1	43	43
2	47	47
3	53	53
4	61	61
5	71	71
6	83	83
7	97	97
8	113	113
9	131	131
10	151	151
11	173	173
12	197	197
13	223	223
14	251	251
15	281	281
16	313	313
17	347	347
18	383	383
19	421	421
20	461	461
21	503	503
22	547	547
23	593	593
24	641	641
25	691	691
26	743	743
27	797	797
28	853	853

The other thing to note is that the semi-primes appear at a very regular intervals .. 2,4,6,8,10……

Once x gets to 2 x 41 (remember the 41 was added on  ( x^2+x+41) .. another sequence starts …

81	6683	41	163
82	6847	41	167
83	7013	7013
84	7181	43	167
85	7351	7351
86	7523	7523
87	7697	43	179
88	7873	7873
89	8051	83	97
90	8231	8231
91	8413	47	179
92	8597	8597
93	8783	8783
94	8971	8971
95	9161	9161
96	9353	47	199
97	9547	9547
98	9743	9743
99	9941	9941
100	10141	10141
101	10343	10343
102	10547	53	199
103	10753	10753
104	10961	97	113
105	11171	11171
106	11383	11383
107	11597	11597
108	11813	11813
109	12031	53	227
110	12251	12251
111	12473	12473
112	12697	12697

This time the factors are made up of those that appeared in the values of x between 1 and 40 and new values that do not occur in the sequence. These can be seen to be 163,167, 179,199 ,227 … and they appear regularly spaced too.

You can try this for yourself at home .. (as they say) and you will note every time X increases by 41 another sequence springs up. I have worked out a formula for the sequence starts and their gaps.
While in the lake district this year I got up early most mornings to work on a general formula for all these sequences and come up with this  ..

(n^2*x^2) +/- ((n^2-2n)*x) + (n^2*k – (n-1)) ………………. (k = to 41)

The table below shows how each sequence follows the above formula:

n	x	k	(n^2*x^2)+((n^2-2n)*x)+(n^2*k-(n-1)
1	0	41	41
1	1	41	43
1	2	41	47
1	3	41	53
2	0	41	163
2	1	41	167
2	2	41	179
2	3	41	199
3	0	41	367
3	1	41	373
3	2	41	397
3	3	41	439
4	0	41	653
4	1	41	661
4	2	41	701
4	3	41	773
5	0	41	1021
5	1	41	1031
5	2	41	1091
5	3	41	1201

I decided to publish this formula against one specific set of results in the OEIS .. It was allocated A214732. and is when n=5 so the generic formula becomes  …. 25x^2 + 15x+1021. It produces around 16 contiguous primes starting at x=0 –> x=16.

So based on the theory that the OEIS will be around long after I’ve shuffled off this mortal coil … this along with the primorial conjecture .. is my bold claim on posterity … ya what I hear you say … get a life ….. :)

Just some stuff that makes primes weird.

Here are some observations that may be of interest. The following list of numbers are the primes from 0 through to 71 in sequence (including 1). If you add 1 to the square of the number you will make a total of 30030.  So what? You may ask and rightly so! Except that 30030 is primorial (13) (1x2x3x5x7x11x13) which is odd, isn’t it?

0           1

1 –>       2

2–>        5

3–>      10

5–>      26

7–>      50

11–>   122

13–>   170

17–>   290

19–>   362

23–>   530

29–>   842

31–>   962

37–> 1370

41–> 1682

43–> 1850

47–> 2210

53–> 2810

59–> 3482

61–> 3722

67–> 4490

71–> 5042

—————-

30030

Also

11^2 + 13^2 + 17^2 + 19^2 +23^2 + 29^2 = 2310 .. which is 2x3x5x7x11 (or primorial(11))

Here is another contiguous prime sequence – this is note able as 199 is the first prime south side of 210 7(p#). You might expect this result if the primes were centered about 105 (p(7#)/2)  but clearly they are not in this case.

199+211+223+227+229+233+239+241+251+257 = 2310 (11p#)

also ( and this, I agree, is slightly more tenuous) but if you sum all the primes between 1 and 59  in the manner shown below
(1^2 + 2^2) + (2^2+3^2) + (3^2 + 5^2) +  ….(47^2+53^2) + (53^2+59^2) = 30032(p#13 + 2)

5
11
17
29
41
59
71
101
107
137
149
179
191
197
227
239
269
281

+

________

2310

This time I’ve added the (6n-1) component of the first 18 twin primes and got to 2310 again ;-)

Just observations that may help someone along the way .
BP 19/11/2009