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