Supplement to "Palindromic Prime Pyramids"
(an article by G. L. Honaker, Jr. and Chris K. Caldwell)

Original article: Adobe Acrobat (38k) | postscript (704k) | Microsoft Word (135k)
(To appear in the Journal of Recreational Mathematics, Baywood Publishing)

 

In the paper "Palindromic prime pyramids" [HC2000] we discuss a type of pyramid first proposed by G. L. Honaker, Jr. For example, starting with the prime 2 we have

2               2
929 929
39293 39293
7392937 3392933
373929373 733929337

Note that each row is a palindromic prime (a palprime) with the previous row as the central digit.  These are the tallest that can be built starting with 2 and with step size one (adding one digit at a time to each side.) 

Step size two

In table 1 of that article we give one of the two tallest (starting with 2) and with step two.  Here is the other:

2
70207
357020753
9635702075369
33963570207536933
723396357020753693327
1272339635702075369332721
97127233963570207536933272179
119712723396357020753693327217911
9011971272339635702075369332721791109
33901197127233963570207536933272179110933
943390119712723396357020753693327217911093349
3894339011971272339635702075369332721791109334983
19389433901197127233963570207536933272179110933498391
151938943390119712723396357020753693327217911093349839151
7515193894339011971272339635702075369332721791109334983915157
74751519389433901197127233963570207536933272179110933498391515747
127475151938943390119712723396357020753693327217911093349839151574721
3012747515193894339011971272339635702075369332721791109334983915157472103
73301274751519389433901197127233963570207536933272179110933498391515747210337
337330127475151938943390119712723396357020753693327217911093349839151574721033733
9933733012747515193894339011971272339635702075369332721791109334983915157472103373399
72993373301274751519389433901197127233963570207536933272179110933498391515747210337339927
927299337330127475151938943390119712723396357020753693327217911093349839151574721033733992729
1892729933733012747515193894339011971272339635702075369332721791109334983915157472103373399272981
13189272993373301274751519389433901197127233963570207536933272179110933498391515747210337339927298131

When finding these we found all palprime pyramids starting with 2 and step size two.  The actual number of pyramids at each level (starting with 2, step size two) is as follows:

height

number

height

number

height

number

height

number

1

1

8

2322

15

2559

22

95

2

9

9

3216

16

1928

23

48

3

42

10

3742

17

1362

24

19

4

136

11

4212

18

867

25

7

5

369

12

4070

19

539

26

2

6

779

13

3753

20

327

 

 

7

1491

14

3142

21

170

 

 

We also have a page with the last row of the longest pyramids with step size two, starting at various other primes.

Step size three is hard!

Starting with 2, there should be one of height around 193 (ending with a 1153 digit prime), but to find the tallest such pyramid by exhaustive search we'd have to check about 1030 different possibilities.  Keeping just 160 pyramids at each step we found two of height 94 rows, each ending with a 559 digit prime.  (Actually we only used probable primality tests.)   Here is the last row of one:

  361119744 7173511717 2375698479 9396993986 7029487207 6673738872 5709759330 3693669459 1471136170 2752966900 9781309039 3814299595 2930303170 7603571863 2439535136 1312378798 3219633967 8074474010 2964920975 7309381331 2017933390 1120921380 1813963023 4276013533 0951701970 1207407113 8190030076 0930330300 2003033039 0670030091 8311704702 1079107159 0335310672 4320369318 1083129021 1093339710 2133183903 7579029469 2010474470 8769336912 3897873213 1631535934 2368175306 7071303039 2595992418 3930903187 9009669257 2071631174 1954966396 3033957907 5278837376 6702784920 7689399693 9974896573 2717115371 7447911163

We also could have ended in this 559 digit probable-prime:

  345387772 1647503541 2677112375 0334308397 7101533331 3312238138 4933951151 1073451997 3297712071 2194199354 9327333123 4576811474 4344924120 9583571863 2439535136 1312378798 3219633967 8074474010 2964920975 7309381331 2017933390 1120921380 1813963023 4276013533 0951701970 1207407113 8190030076 0930330300 2003033039 0670030091 8311704702 1079107159 0335310672 4320369318 1083129021 1093339710 2133183903 7579029469 2010474470 8769336912 3897873213 1631535934 2368175385 9021429443 4474118675 4321333723 9453991491 2170217792 3799154370 1151159339 4831832213 3133335101 7793803433 0573211776 2145305746 1277783543

We tested using prp tests and the 160 limit for starting values of 2, 3, 5 and 7 (step size 3). Here it the number we found  (2=diamond, 3=cross, 5=circle, 7=box) compared to the estimate (solid curve).

 

We have limited data for other starting values.

Other notes

We noted C. Rivera found 164 terms (see http://www.ping.be/~ping6758/palprim3.htm) of the "condensed sequence" [A053600]. He adds:

For testing the primality of the numbers of the pyramid I used a "strong-pseudo prime" test and code, according to the pseudocode (Algorithm 6.1) described in the David M. Bressoud's book [Bressoud89] "Factorization & primality testing", Springer-Verlag, p. 77, running under UBASIC.

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