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CONTENTS

Composite, Prime

Palindromic Sophie Germaine Primes (1 & 2)

Prime 30103

Fermat & Mersenne Numbers

Minimum Difference Prime Squares Prime Queen Problem

Ulam's Prime Spiral

Primes Adjacent to 6n

A Reversible Sequence

Prime 5882353

Primes Plus Even Number 8 Consecutive Primes

Prime Rectangles

A Prime Staircase

Prime 1129

Fermat Product Plus 2

Perfect Numbers from Primes

Primeval Primes

A Prime Circle

Prime # 57

Butterfly Primes

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, 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 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 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.

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 x2 + 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   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

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 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 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 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.

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

π5757 - 24 is prime

57 x 232 - 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 2n ) 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.brooksdesign-ps.net/reginald_brooks/code/html/netart75.htm

This page was originally posted December 1998
It was last updated June 04, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz