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)
191 &
383
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,
pp3841t
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 multidigit 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 forwards). 
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 
Prime Queen Problem
Proposed by G. L. Honaker, Jr.
on Nov. 15, 1998.
37 
24 
45 
4 
39 
22 
47 
62 
44 
5 
38 
23 
46 
61 
40 
21 
25 
36 
43 
60 
3 
20 
63 
48 
6 
59 
26 
35 
64 
41 
2 
19 
27 
30 
57 
42 
1 
34 
49 
12 
58 
7 
54 
29 
52 
13 
18 
15 
31 
28 
9 
56 
33 
16 
11 
50 
8 
55 
32 
53 
10 
51 
14 
17 

Find the greatest number of prime squares that
a queen can attack if placed on an n by n
knight's tour solution. For the purpose of this problem, when
considering if the queen is attacking a particular square, assume
the intervening squares are vacant.
The knight's tour is a numbered tour of a knight over a otherwise
empty board visiting each square once only.
The possible tour solutions are in the trillions.
3 
14 
19 
24 
1 
20 
9 
2 
13 
18 
15 
4 
25 
8 
23 
10 
21 
6 
17 
12 
5 
16 
11 
22 
7 

By early December, 1998, Mike Keith had found these two solutions.. The
order8 has all of the 18 prime numbers less then 64 under attack by the queen
placed on number 35. The order5 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 order6 and order7
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 order9 or greater.
Can you find any other perfect solutions to this problem?
Addendum On April 1, 2004, I received an email
from Jacques Tramu. He enclosed a solution he had found for the perfect
order 9 square of this type. All 22 primes in the number range of 1 to 81
are attacked by the queen.
He shows the solution at
http://mapage.noos.fr/echolalie/q9.htm
A few days later I also received a notice of this solution from G. L.
Honaker, Jr., the originator of this problem. 
13 
76 
15 
20 
11 
74 
25 
22 
9 
16 
19 
12 
75 
26 
21 
10 
67 
24 
77 
14 
17 
28 
73 
56 
23 
8 
69 
18 
81 
78 
43 
60 
27 
68 
57 
66 
79 
44 
29 
2 
55 
72 
59 
70 
7 
30 
51 
80 
61 
42 
3 
36 
65 
58 
45 
48 
41 
52 
1 
54 
71 
6 
35 
50 
31 
46 
39 
62 
33 
4 
37 
64 
47 
40 
49 
32 
53 
38 
63 
34 
5 

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
272 
213 
214 
215 
216 
217 
218 
219 
220 
221 
222 
223 
224 
225 
226 
227 
271 
212 
161 
162 
163 
164 
165 
166 
167 
168 
169 
170 
171 
172 
173 
228 
270 
211 
160 
117 
118 
119 
120 
121 
122 
123 
124 
125 
126 
127 
174 
229 
269 
210 
159 
116 
81 
82 
83 
84 
85 
86 
87 
88 
89 
128 
175 
230 
268 
209 
158 
115 
80 
53 
54 
55 
56 
57 
58 
59 
90 
129 
176 
231 
267 
208 
157 
114 
79 
52 
33 
34 
35 
36 
37 
60 
91 
130 
177 
232 
266 
207 
156 
113 
78 
51 
32 
21 
22 
23 
38 
61 
92 
131 
178 
233 
265 
206 
155 
112 
77 
50 
31 
20 
17 
24 
39 
62 
93 
132 
179 
234 
264 
205 
154 
111 
76 
49 
30 
19 
18 
25 
40 
63 
94 
133 
180 
235 
263 
204 
153 
110 
75 
48 
29 
28 
27 
26 
41 
64 
95 
134 
181 
236 
262 
203 
152 
109 
74 
47 
46 
45 
44 
43 
42 
65 
96 
135 
182 
237 
261 
202 
151 
108 
73 
72 
71 
70 
69 
68 
67 
66 
97 
136 
183 
238 
260 
201 
150 
107 
106 
105 
104 
103 
102 
101 
100 
99 
98 
137 
184 
239 
259 
200 
149 
148 
147 
146 
145 
144 
143 
142 
141 
140 
139 
138 
185 
240 
258 
199 
198 
197 
196 
195 
194 
193 
192 
191 
190 
189 
188 
187 
186 
241 
257 
256 
255 
254 
253 
252 
251 
250 
249 
248 
247 
246 
245 
244 
243 
242 
Prime spiral generated with Euler's formula x^{2}
+ x + 17
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 x^{2} + 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). 

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, 19992000, pp.
246250. A short introductory article appeared in JRM 29:3, pp.
188189
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 
Dr. Ellerstein pointed out that
Ulam's spiral also has the diagonal of squares with the other
diagonal consisting of pronics (as does his).
Here I show Ulam's spiral starting at 0. But
instead of showing the primes, I show the diagonal of squares and
the diagonal of pronics.
Pronics are the product of two adjacent numbers. 
72
71 70 69 68 67 66 65 64 99
73 42 41 40 39 38 37
36 63 98
74 43 20 19 18 17
16 35 62 97
75 44 21 06 05
04 15 34 61 96
76 45 22 07 00 03 14 33 60 95
77 46 23 08 01 02 13 32 29
94
78 47 24 09 10 11
12 31 58 93
79 48 25 26 27 28 29
30 57 92
80 49 50 51 52 53 54 55
56 91
81 82 83 84 85 86 87 88 89
90 
A prime spiral arranged as a circle is thoroughly discussed and
illustrated at Rom Sacks
http://www.numberspiral.com/
Primes Adjacent to 6n
Is there at least 1 prime adjacent to each
multiple of 6? 
Addendum Nov. 5,
2002 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 12^{2}  5^{2}, 120 = 6n, and 121 = 11^{2} 
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 =
588^{2} + 2353^{2}
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 Plus Even
Number
Does this pattern continue producing primes forever?
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, p 204
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 emailed 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://wwwgroups.dcs.stand.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.
A Prime Circle
I received this diagram via email from a C. Robin in
May of 2003. Notice that along each radial, the numbers are
separated by 24.
Click on the image to
enlarge it. 

Prime # 57
π57^{57}
 2^{4} is prime
57 x 2^{32}  1 is prime
57 = sum of 3 primes = 5 + 23 + 29
57 = sum of the first 4 square # = 1 + 4 + 16 + 36
57 + (100 x 2^{n} ) are primes for n = 1, 0, 1, 2, 3,
4 & 5
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.brooksdesignps.net/reginald_brooks/code/html/netart75.htm
