More Prime Patterns

CONTENTS

Composite, Prime

NOTE: pattern alternates between composite and prime (until x=10). Too bad !

A. H. Beiler, Recreations in the Theory of Numbers p. 85

Palindromic Sophie Germaine Primes (1)

39493939493 & 78987878987

If P is greater than 2 and is a prime, than if 2P+1 is also prime, P is known as a Sophie Germaine prime. There are many such primes but only 71  pairs with three to eleven digits if  both primes are palindromic. Above are the lowest and the highest such pairs.

If Q (2P+1) is itself a Sophie Germaine prime there are a total of 19 such triplets where each prime is palindromic. They range in size from 23 digits long to 39 digits long The smallest such triplet follows:

19091918181818181919091,     38183836363636363838183,      76367672727272727676367

Harvey Dubner, JRM 26:1, 1994, pp38-41t

Palindromic Sophie Germaine Primes (2)

If we let A=1, B=2, C=3, etc than

P+A+L+I+N+D+R+O+M+E+S + A+R+E + F+U+N = 191

Also

15551 (the smallest palindromic prime producing another in this way) when written in words

F+I+F+T+E+E+N + T+H+O+U+S+A+N+D +
F+I+V+E + H+U+N+D+R+E+D + F+I+F+T+Y + O+N+E = 383

This courtesy of G. L. Honaker, Jr. (See Prime Queen Problem below)

Prime 30103

30103 is the only known multi-digit palindromic prime found by averaging the divisors of a composite number.

30103 = (1 + 5 + 173 + 865 + 29929 + 149645)/6
30103 = (1 + 2 + 3 + 6 + 173 + 346 + 519 + 1038 + 29929 + 59858 + 89787 + 179574)/12

30103 = average of divisors of 149645
30103 = average of divisors of 179574

It was found by Jud McCranie and G. L. Honaker, Jr. in July/98.

See Carlos Rivera’s Prime Puzzles & Problems

Fermat & Mersenne Numbers

Fermat numbers , when expressed in binary, have all zeros with a one at each end. Mersenne numbers have all ones.
Both the decimal and binary numbers are palindromes (read the same backwards as forewards).

Minimum Difference Prime Squares

These two squares each contain the 25 primes that are less then 100.

Add
The maximum sum of any row, column or diagonal is 213
The minimum sum is 211
The difference (which is the minimum possible) is 2

Multiply
The maximum product of any line, column or diagonal is 19013871
The minimum product is 18489527
The difference which is also the minimum possible .is 524344

JRM vol. 26: 4, 1994 ,p. 308,309 Prob. 2094 proposed by R. M. Kurchan, solution by M. Reed

By early December, 1998, Mike Keith had found these two solutions.. The order-8 has all of the 18 prime numbers less then 64 under attack by the queen placed on number 35. The order-5 square is also perfect because the queen is attacking all 9 of the primes when placed on the number 25.

In fact, Mr. Keith found perfect solutions also for order-6 and order-7 squares. It is impossible to have a knight tour solution for orders 2, 3 , or 4 and Mr. Keith conjectures that it is impossible to have a perfect solution for boards of order-9 or greater.

Addendum:

Addendum2:
April 12, 2004. 5 minutes after uploading the previous addendum, I checked my email. Another update from G.L.
Jacques has just published a perfect order 10! See it at his link shown above.

See Mike Keith's information on this problem at http://users.aol.com/s6sj7gt/primeq.htm

Ulam's Prime Spiral

Probably everyone has heard the story about how Stanislaw M. Ulam discovered this pattern while sitting in a scientific meeting listening to what he described as a "long and boring paper." My first experience with his prime spiral was in the mid 70’s when I programmed it on my Apple II computer using Euler’s formula   x2 + x + 41. The graphics screen had a resolution of 40 x 40, and I can remember the thrill I received as I watched the screen slowly (but in those days I thought it was fast) fill up with a solid diagonal of 40 primes from one corner to the other. A good explanation of this pattern is in M. Gardner, Sixth Book of Mathematical Games, Scribner’s, 1971 The triangle plot and the linear plot both show the primes in neat columns. The linear plot with six numbers per line illustrates how all primes are 6n - 1 or 6n + 1 (except for 2 and 3).

The Ellerstein Spiral

Dr. Stuart Ellerstein expanded on this theme in a paper that appeared as The Pronic Renaisance II, in The Journal of Recreational Mathematics, Vol. 30:4, 1999-2000, pp. 246-250. A short introductory article appeared in JRM 29:3, pp. 188-189 The following image is from that paper. Note particularly that the even squares, and the odd squares - 1 appear in the right diagonal of this array. Note: A pronic number is one with the form x(x+1). It is also twice a triangular number.

Click for larger view

A prime spiral arranged as a circle is thoroughly discussed and illustrated at Rom Sacks  http://www.numberspiral.com/

Primes Adjacent to 6n

Dr. Ellerstein (see above section) advised me that the answer to the above question is NO!
The first exception is for n = 20 
The 3 adjacent numbers are 119, 120, and 121    (119 = 7 x 17 and also 12² - 5², 120 = 6n, and 121 = 11²

A Reversible Sequence

1193, 1201, 1213, 1217,1223,1229,1231, 1237, 1249, 1259

A sequence of ten consecutive primes. Each one of these primes is still a prime when the order or the digits is reversed.

This sequence was discovered by Carlos Rivera, and is registered by him in Sloane’s Integer Sequences as A040104.

Prime 5882353

5882353 = 588² + 2353²

This is the only prime number I know with this property. Are there other primes like this?

See examples of composite numbers with this or similar properties at my Narcissistic Numbers page.

Primes + Even Number

Addendum Nov. 19, 2002
No. See Ulam's Spirol (above). Aguydude pointed this out to me.

8 Consecutive Primes

8 consecutive primes 11    13   17   19   23   29   31    37
the difference between   2     4     2      4      6     2      6

These eight consecutive primes have the form k, k + 2, k + 6, k + 8, k + 12, k + 18, k + 20, k + 26.

The next such series starts with the prime 15,760,091 and  there are 6 more such series below 1,000,000,000

Martin Gardner, The Last Recreations, p204

Prime Rectangles

This pattern has 8 rows of 11 consecutive primes summing to a prime number.
Each row stars with the second prime in the row above it.
To save space, all primes (except the totals) are of the following form: 352XXXX

This pattern was found by Carlos Rivera and appeared on Prime Puzzles & Problems in Aug./98.

Since then, Jud McCranie has found many more of these patterns. See his pattern with 9 rows of 97 primes each at
Carlos Rivera’s WWW site .

A Prime Staircase

To save space, all primes (except the totals) are of the following form: 12606XXXX

There are three more lines to this pattern. The first and last primes in the last line are 126064739 and 126065033. The sum of this line is 2143103093.

This pattern was found by Carlos Rivera and appeared on his Prime Puzzles & Problems in Aug./98.

Prime 1129

Primes 1009 and 1201 have the same property. The smaller prime numbers 241,409, 601, and 769 almost have this property. In each case the only n for which there is no solution is n = 7. Charles Ashbacher, JRM 24:3, 1992, pp202 Carlos Rivera (see his Prime Puzzles & Problems) sent me on Aug. 8/98, the solution below for n equals the first ten even numbers. Two days later he e-mailed me the first example he had found where n is restricted to the first ten odd numbers.

Fermat Product Plus 2

Every Fermat number is the product of all previous Fermat numbers plus 2.

Perfect Numbers from Primes

This is spelled out by Euclid in his Proposition 36 of Book IX of the Elements.

December, 2005: V.V. Raman reports "that every perfect number is 1 (mod 9)?"

See an excellent paper on Perfect numbers at
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfectnumbers.html

Primeval Primes

In each case, the number in the left column is the lowest number that includes this many primes.
The underlined numbers are Primeval Primes.

See more on Primeval Primes at Mike Keith’s page at http:users.//aol.com/s6sj7gt/primeval.htm

A Prime Circle

I received this diagram via email from a C. Robin in May of 2003. Now, in March, 2005 I have finally got around to posting it! Notice that along each radial, the numbers are separated by 24.

Pattern by C. Robin Click on the image to enlarge it.

Prime # 57

(57) = 2⁴ and 57⁵⁷ - 2⁴ is prime

57 x 2²⁵ - 1 is prime

57 + (100 x 2ⁿ ) are primes for n = -1, 0, 1, 2, 3, 4 & 5

I modestly refer to this as "Heinz's number".

See more curios at http://primes.utm.edu/curios/

Butterfly Primes

Reginald Brooks has recently completed an extensive work on what he refers to as Butterfly Primes.

I was made aware of this by several emails from him, the most recent being March 2, 2006.

His site contains much information and many attractive multicolored tables.

This is a prime example of what I (and others) have often claimed, that 
"mathematics is a study of the patterns of numbers!"

Reginald's papers (including other number pattern topics) may be viewed at

http://www.brooksdesign-cg.com/Code/Html/arthry2.htm

Please send me Feedback about my Web site!

Last updated June 19, 2007
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998, 2000 by Harvey D. Heinz