Fw: Last digits of prime numbers

[Richard Donovan]

Sent from Outlook for iOS

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From: Richard Donovan <rsdo...@hotmail.com>
Sent: Friday, April 17, 2026 11:27:09 AM
To: J Programming <progr...@jsoftware.com>
Subject: Last digits of prime numbers

 

Hi

There has been a recent Numberfile video re new investigations into

the sequence of the last digits of prime numbers. It has  been know for

Some time that the possible candidates (1, 3, 7, 9) are not present in

Equable amounts, and there has recently been a new discovery which

Supposes that certain of the prime endings have a bias to what the next

Ending might be.

In order to follow this discovery I have written the following J program that

Runs through a given number of primes and tabulates the amount of prime

Endings. This is the first step before I consider biases in following primes.

This is the program I am using:

1#.'1379'=/>{.each |.each ":each p:i.1000000

249934 250110 250014 249940

It seems to produce the correct results but I am sure there must be a

Better (and no doubt faster) way to do this calculation.

I'm sure I don't need three 'eachs' but can't find a way round it.

Grateful for any help,

Thanks in advance,

Richard

[chris burke]

(#,{.) /.~ 2 5 -.~ 10 | p: i.1000000
250110 3
250014 7
249934 1
249940 9

[Richard Donovan]

Thanks!

Sent from Outlook for iOS

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From: chris burke <cbu...@jsoftware.com>
Sent: Friday, April 17, 2026 3:01:37 PM
To: fo...@jsoftware.com <fo...@jsoftware.com>
Subject: Re: [Jforum] Fw: Last digits of prime numbers

 

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[Jan-Pieter Jacobs]

I was about to suggest using (,#)/..~ rather than ({.,#)/.~ but was surprised to see it's far slower and fatter, while I thought this was exactly the use-case for which /.. was introduced...

100 timespacex '({.,#)/.~ 2 5-.~ 10 | pr'

0.0046416 8.39088e6

100 timespacex '(,#)/..~ 2 5-.~ 10 | pr'

0.0113772 1.67963e7

Jan-Pieter

[Henry Rich]

({., #)/. Is backed by special code. (,#)/.. should use the same, I guess. 

Henry Rich

[Remington Furman]

It has been know for Some time that the possible candidates (1, 3, 7, 9) are not present in
Equable amounts,

Not true in base 2, where every prime after 2 ends in 1.  ;)

   (] /: {."1)&.>  <@(({.,#)/.~)"(1) (2+i.15) |"(0 1) p: i.1000000
+--------+--------+--------+--------+--------+--------+--------+--------+--------+---------+---------+--------+---------+---------+---------+
|0      1|0      1|1 499798|0      1|1 499829|0      1|1 249796|1 166568|1 249934| 0      1| 1 249739| 0     1| 1 166671| 1 124840| 1 124858|
|1 999999|1 499829|2      1|1 249934|2      1|1 166671|2      1|2 166751|2      1| 1  99925| 2      1| 1 83314| 2      1| 2 124960| 2      1|
|        |2 500170|3 500201|2 250015|3      1|2 166651|3 250091|3      1|3 250110| 2 100000| 3      1| 2 83341| 3 166708| 3      1| 3 125116|
|        |        |        |3 250110|5 500169|3 166708|5 250002|4 166637|5      1| 3 100051| 5 250059| 3 83355| 5 166683| 4 124890| 5 124931|
|        |        |        |4 249940|        |4 166636|7 250110|5 166707|7 250014| 4  99994| 7 250090| 4 83267| 7      1| 5      1| 7 125001|
|        |        |        |        |        |5 166683|        |7 166624|9 249940| 5 100007|11 250110| 5 83396| 9 166650| 7 125055| 9 124938|
|        |        |        |        |        |6 166650|        |8 166712|        | 6 100082|         | 6 83369|11 166636| 8 125065|11 124975|
|        |        |        |        |        |        |        |        |        | 7  99983|         | 7 83367|13 166650|11 125094|13 125071|
|        |        |        |        |        |        |        |        |        | 8  99905|         | 8 83339|         |13 125044|15 125109|
|        |        |        |        |        |        |        |        |        | 9 100024|         | 9 83309|         |14 125050|         |
|        |        |        |        |        |        |        |        |        |10 100028|         |10 83213|         |         |         |
|        |        |        |        |        |        |        |        |        |         |         |11 83412|         |         |         |
|        |        |        |        |        |        |        |        |        |         |         |12 83317|         |         |         |
+--------+--------+--------+--------+--------+--------+--------+--------+--------+---------+---------+--------+---------+---------+---------+

-Remington

[Richard Donovan]

It has also been suggest that some digits are more likely

follow other digits than seems random.

EG 1 follow 9 more that 7 follows 3 (as consecutive last

digits than would be expected.

I am trying to make a matrix that shows the percentage of y that

follows x such as

          d1         d3          d7         d9    

d1    %d1d1 %d1d3 %d1d7 %d1d9

d3    %d3d1

d7

d9

where %d1d3 is %age of times 3 follows 1 as last digit of prime etc

If I ever get this working I guess it would be interesting to investigate

the third number in the series etc!

Richard