Puzzle 1038. Three puzzles about consecutive primes.

Problems & Puzzles: Puzzles

Puzzle 1038. Three puzzles about consecutive primes.

Paolo Lava sent the following nice puzzle:

a) At which exponent we can arise 2 and get a number with two consecutive primes in its decimal representation?

The first ones for increasing value of primes are:

41 and 43 in 2^22 = 4194304

647 and 653 in 2^124 = 21267647932558653966460912964485513216

1987 and 1993 in 2^212 = 6582018229284824168619876730229402019930943462534319453394436096

2213 and 2221 in 2^196 = 100433627766186892221372630771322662657637687111424552206336

Next ones?

Of course, we can repeat the search for any integer > 2.

E.g. the first ones for different 2<n<26 are:

n

exponent

P1

P2

3

69

331

337

4

11

41

43

5

121

2549

2551

6

8

61

67

7

19

31

37

8

9

13

17

9

35

499

503

10

N.A.

N.A.

N.A.

11

26

181

191

12

33

101

103

13

13

53

59

14

9

61

67

15

6

11

13

16

9

71

73

17

12

23

29

18

2

2

3

19

7

71

73

20

22

41

42

21

6

5

7

22

4

3

5

23

1

2

3

24

2

5

7

25

47

977

983

Even more interesting when the power produces 3 consecutives primes. For instance:

8^31 = 9903520314283042199192993792 -> 919, 929, 937;

14^13 = 793714773254144 -> 71, 73, 79;

17^19 = 239072435685151324847153 -> 43, 47, 53; (by the way, this is even more interesting because 17 and 19 are themselves consecutive primes).

22^6 = 113379904 -> 7, 11, 13.

Other values:

36^25 = 808281277464764060643139600456536293376 -> 643, 647, 653;

56^14 = 2982856619293778479415296 -> 929, 937, 941;

69^3 = 328509 -> 2, 3, 5;

73^22 = 98424433237708439716398638596388483974129 -> 433, 439, 443;

76^33 = 116640277783837151685634195581910452303930145165533737937534976 -> 373, 379, 383;

81^38 = 3329896365316142756322307042065269797678257903507506764336659647715734241 -> 643, 647, 653;

85^12 = 142241757136172119140625 -> 11, 13, 17;

91^3 = 753571 -> 3, 5, 7;

(36^25 and 81^38 generate the same triplet 643, 647, 653)

b) The product of two consecutive primes has the next one in its representation

p(n)*p(n+1)    ->   p(n+2)

17*19 = 232 -> 23

499*503 = 250997 -> 509

1997*1999 = 3992003 -> 2003

4999*5003 = 25009997 -> 5009

9791*9803 = 95981173 -> 9811

5999993*6000011 = 36000023999923 -> 6000023

480083407*480083419 = 230480083437728533 -> 480083437

600000019*600000041 = 360000036000000779-> 600000077

and viceversa

c) The product of two consecutive primes has the previous one in its representation

p(n+1)*p(n+2)    ->   p(n)

5*7 = 35 -> 3

373*379 = 141367 -> 367

1999993*2000003 = 3999991999979 -> 1999979

37015607*37015621 = 1370155679796947 -> 37015567

Q. Can you get other solutions for each of the three issues posted by Paolo?

During the week 8-14, May 2021, contributions came from: Giorgos Kalogeropoulos, Shyam Sunder Gupta, Paul Cleary, Metin Sariyar, Adam Stinchcombe, Hakan Summakoglu, Simon Cavegn, Oscar Volpatti and Emmanuel Vantieghem

***

Giorgos wrote:

a) Here is an example with 8-digits consecutive primes
2^200000 -> 25687507 -> 25687511

The following chart is extended and updated with 7-digits consecutive primes

n^exponent->Prime1 -> Prime2

2^20000 -> 6685307 -> 6685331
3^20000 -> 1924457 -> 1924459
4^10000 -> 6685307 -> 6685331
5^9000 -> 6887147 -> 6887149
6^8000 -> 9695633 -> 9695669
7^7502 -> 4393637 -> 4393643
8^7000 -> 1778531 -> 1778537
9^10000 -> 1924457 -> 1924459
11^8015 -> 1140449 -> 1140463
12^8001 -> 3724099 -> 3724103
13^8019 -> 1001569 -> 1001587
14^6000 -> 3724907 -> 3724913
15^7000 -> 7728571 -> 7728599
16^6000 -> 6299819 -> 6299831
17^6001 -> 6534943 -> 6535019
18^6003 -> 5531107 -> 5531117
19^7000 -> 2586719 -> 2586721
20^20000 -> 6685307 -> 6685331
21^5000 -> 2927581 -> 2927591
22^5003 -> 6563663 -> 6563677
23^4504 -> 5203171 -> 5203181
24^4506 -> 7256681 -> 7256693
25^4000 -> 2284223 -> 2284229
26^4011 -> 8112971 -> 8112977
27^4001 -> 4970591 -> 4970611
28^4001 -> 5680501 -> 5680559
29^4002 -> 2427769 -> 2427773
30^13002 -> 9479801 -> 9479809

Here are two examples with 3 consecutive 7-digits primes

2^500000 ->3390073 -> 3390083 -> 3390091

2^500000 ->6763613 -> 6763621 -> 6763649

b) p(n) * p(n+1)    ->   p(n+2)

6666533329 * 6666533333 = 44442666653333955557 -> 6666533339

c) p(n+1) * p(n+2)    ->   p(n)

  519794713 * 519794729 = 270186551979467777 -> 519794677

2791000027 * 2791000073 = 7789681279100001971 -> 2791000019

  5236000027 * 5236000073 = 27415696523600001971 -> 5236000019

  5593000027 * 5593000073 = 31281649559300001971 -> 5593000019

5656000027 * 5656000073 = 31990336565600001971 -> 5656000019

  6256000027 * 6256000073 = 39137536625600001971 -> 6256000019

  7170000023 * 7170000077 = 51408900717000001771 -> 7170000017

  7533000023 * 7533000077 = 56746089753300001771 -> 7533000017

8996000023 * 8996000077 = 80928016899600001771 -> 8996000017

***

Shyam wrote:

I have obtained the results of this puzzle for Question 2 and 3 for all values of primes < 10^10

b) The product of two consecutive primes has the next one in its representation

p(n)*p(n+1)->p(n+2)

17*19=232->23

499*503=250997->509

1997*1999=3992003->2003

4999*5003=25009997->5009

9791*9803=95981173->9811

5999993*6000011=36000023999923->6000023

480083407*480083419=230480083437728533->480083437

600000019*600000041=360000036000000779->600000077

6666533329*6666533333=44442666653333955557 ->6666533339

c) The product of two consecutive primes has the previous one in its representation

p(n+1)*p(n+2)->p(n)

5*7=35->3

373*379=141367->367

1999993*2000003=3999991999979->1999979

37015607*37015621=1370155679796947->37015567

519794713 * 519794729 = 270186551979467777 -> 519794677

2791000027 * 2791000073 = 7789681279100001971 -> 2791000019

5236000027 * 5236000073 = 27415696523600001971 -> 5236000019

5593000027 * 5593000073 = 31281649559300001971 -> 5593000019

5656000027 * 5656000073 = 31990336565600001971 -> 5656000019

6256000027 * 6256000073 = 39137536625600001971 -> 6256000019

7170000023 * 7170000077 = 51408900717000001771 -> 7170000017

7533000023 * 7533000077 = 56746089753300001771 -> 7533000017

8996000023 * 8996000077 = 80928016899600001771 -> 8996000017

***

Paul wrote:

a).

It seemed quite easy to find solutions with 2 consecutive 2 digit primes and after extending the search from 3 consecutive to 4 and so on I decided to head straight to the top with all 21 x 2 digit primes. Due to large numbers and many of them here are just the n and exponent and the digit length from n = 2 to 199.

2^563 Contains all 21 consecutive 2 digit primes with a Digit length of 170

3^317 Contains all 21 consecutive 2 digit primes with a Digit length of 152

4^312 Contains all 21 consecutive 2 digit primes with a Digit length of 188

5^330 Contains all 21 consecutive 2 digit primes with a Digit length of 231

6^261 Contains all 21 consecutive 2 digit primes with a Digit length of 204

7^190 Contains all 21 consecutive 2 digit primes with a Digit length of 161

8^208 Contains all 21 consecutive 2 digit primes with a Digit length of 188

9^195 Contains all 21 consecutive 2 digit primes with a Digit length of 187

11^207 Contains all 21 consecutive 2 digit primes with a Digit length of 216

12^191 Contains all 21 consecutive 2 digit primes with a Digit length of 207

13^213 Contains all 21 consecutive 2 digit primes with a Digit length of 238

14^150 Contains all 21 consecutive 2 digit primes with a Digit length of 172

15^169 Contains all 21 consecutive 2 digit primes with a Digit length of 199

16^156 Contains all 21 consecutive 2 digit primes with a Digit length of 188

17^154 Contains all 21 consecutive 2 digit primes with a Digit length of 190

18^166 Contains all 21 consecutive 2 digit primes with a Digit length of 209

19^156 Contains all 21 consecutive 2 digit primes with a Digit length of 200

20^563 Contains all 21 consecutive 2 digit primes with a Digit length of 733

21^161 Contains all 21 consecutive 2 digit primes with a Digit length of 213

22^123 Contains all 21 consecutive 2 digit primes with a Digit length of 166

23^150 Contains all 21 consecutive 2 digit primes with a Digit length of 205

24^128 Contains all 21 consecutive 2 digit primes with a Digit length of 177

25^165 Contains all 21 consecutive 2 digit primes with a Digit length of 231

26^142 Contains all 21 consecutive 2 digit primes with a Digit length of 201

27^130 Contains all 21 consecutive 2 digit primes with a Digit length of 187

28^111 Contains all 21 consecutive 2 digit primes with a Digit length of 161

29^173 Contains all 21 consecutive 2 digit primes with a Digit length of 253

30^317 Contains all 21 consecutive 2 digit primes with a Digit length of 469

31^131 Contains all 21 consecutive 2 digit primes with a Digit length of 196

32^145 Contains all 21 consecutive 2 digit primes with a Digit length of 219

33^102 Contains all 21 consecutive 2 digit primes with a Digit length of 155

34^122 Contains all 21 consecutive 2 digit primes with a Digit length of 187

35^118 Contains all 21 consecutive 2 digit primes with a Digit length of 183

36^145 Contains all 21 consecutive 2 digit primes with a Digit length of 226

37^134 Contains all 21 consecutive 2 digit primes with a Digit length of 211

38^90 Contains all 21 consecutive 2 digit primes with a Digit length of 143

39^149 Contains all 21 consecutive 2 digit primes with a Digit length of 238

40^312 Contains all 21 consecutive 2 digit primes with a Digit length of 500

41^83 Contains all 21 consecutive 2 digit primes with a Digit length of 134

42^113 Contains all 21 consecutive 2 digit primes with a Digit length of 184

43^145 Contains all 21 consecutive 2 digit primes with a Digit length of 237

44^108 Contains all 21 consecutive 2 digit primes with a Digit length of 178

45^123 Contains all 21 consecutive 2 digit primes with a Digit length of 204

46^126 Contains all 21 consecutive 2 digit primes with a Digit length of 210

47^114 Contains all 21 consecutive 2 digit primes with a Digit length of 191

48^95 Contains all 21 consecutive 2 digit primes with a Digit length of 160

49^95 Contains all 21 consecutive 2 digit primes with a Digit length of 161

50^330 Contains all 21 consecutive 2 digit primes with a Digit length of 561

51^116 Contains all 21 consecutive 2 digit primes with a Digit length of 199

52^155 Contains all 21 consecutive 2 digit primes with a Digit length of 266

53^118 Contains all 21 consecutive 2 digit primes with a Digit length of 204

54^107 Contains all 21 consecutive 2 digit primes with a Digit length of 186

55^111 Contains all 21 consecutive 2 digit primes with a Digit length of 194

56^121 Contains all 21 consecutive 2 digit primes with a Digit length of 212

57^113 Contains all 21 consecutive 2 digit primes with a Digit length of 199

58^106 Contains all 21 consecutive 2 digit primes with a Digit length of 187

59^139 Contains all 21 consecutive 2 digit primes with a Digit length of 247

60^261 Contains all 21 consecutive 2 digit primes with a Digit length of 465

61^121 Contains all 21 consecutive 2 digit primes with a Digit length of 217

62^106 Contains all 21 consecutive 2 digit primes with a Digit length of 190

63^111 Contains all 21 consecutive 2 digit primes with a Digit length of 200

64^104 Contains all 21 consecutive 2 digit primes with a Digit length of 188

65^113 Contains all 21 consecutive 2 digit primes with a Digit length of 205

66^109 Contains all 21 consecutive 2 digit primes with a Digit length of 199

67^119 Contains all 21 consecutive 2 digit primes with a Digit length of 218

68^114 Contains all 21 consecutive 2 digit primes with a Digit length of 209

69^122 Contains all 21 consecutive 2 digit primes with a Digit length of 225

70^190 Contains all 21 consecutive 2 digit primes with a Digit length of 351

71^108 Contains all 21 consecutive 2 digit primes with a Digit length of 200

72^120 Contains all 21 consecutive 2 digit primes with a Digit length of 223

73^98 Contains all 21 consecutive 2 digit primes with a Digit length of 183

74^133 Contains all 21 consecutive 2 digit primes with a Digit length of 249

75^91 Contains all 21 consecutive 2 digit primes with a Digit length of 171

76^118 Contains all 21 consecutive 2 digit primes with a Digit length of 222

77^120 Contains all 21 consecutive 2 digit primes with a Digit length of 227

78^112 Contains all 21 consecutive 2 digit primes with a Digit length of 212

79^83 Contains all 21 consecutive 2 digit primes with a Digit length of 158

80^208 Contains all 21 consecutive 2 digit primes with a Digit length of 396

81^105 Contains all 21 consecutive 2 digit primes with a Digit length of 201

82^89 Contains all 21 consecutive 2 digit primes with a Digit length of 171

83^119 Contains all 21 consecutive 2 digit primes with a Digit length of 229

84^121 Contains all 21 consecutive 2 digit primes with a Digit length of 233

85^120 Contains all 21 consecutive 2 digit primes with a Digit length of 232

86^103 Contains all 21 consecutive 2 digit primes with a Digit length of 200

87^108 Contains all 21 consecutive 2 digit primes with a Digit length of 210

88^108 Contains all 21 consecutive 2 digit primes with a Digit length of 211

89^104 Contains all 21 consecutive 2 digit primes with a Digit length of 203

90^195 Contains all 21 consecutive 2 digit primes with a Digit length of 382

91^145 Contains all 21 consecutive 2 digit primes with a Digit length of 285

92^105 Contains all 21 consecutive 2 digit primes with a Digit length of 207

93^106 Contains all 21 consecutive 2 digit primes with a Digit length of 209

94^112 Contains all 21 consecutive 2 digit primes with a Digit length of 221

95^91 Contains all 21 consecutive 2 digit primes with a Digit length of 180

96^72 Contains all 21 consecutive 2 digit primes with a Digit length of 143

97^113 Contains all 21 consecutive 2 digit primes with a Digit length of 225

98^101 Contains all 21 consecutive 2 digit primes with a Digit length of 202

99^86 Contains all 21 consecutive 2 digit primes with a Digit length of 172

101^106 Contains all 21 consecutive 2 digit primes with a Digit length of 213

102^115 Contains all 21 consecutive 2 digit primes with a Digit length of 231

103^89 Contains all 21 consecutive 2 digit primes with a Digit length of 180

104^102 Contains all 21 consecutive 2 digit primes with a Digit length of 206

105^106 Contains all 21 consecutive 2 digit primes with a Digit length of 215

106^107 Contains all 21 consecutive 2 digit primes with a Digit length of 217

107^88 Contains all 21 consecutive 2 digit primes with a Digit length of 179

108^100 Contains all 21 consecutive 2 digit primes with a Digit length of 204

109^92 Contains all 21 consecutive 2 digit primes with a Digit length of 188

110^207 Contains all 21 consecutive 2 digit primes with a Digit length of 423

111^105 Contains all 21 consecutive 2 digit primes with a Digit length of 215

112^111 Contains all 21 consecutive 2 digit primes with a Digit length of 228

113^109 Contains all 21 consecutive 2 digit primes with a Digit length of 224

114^119 Contains all 21 consecutive 2 digit primes with a Digit length of 245

115^72 Contains all 21 consecutive 2 digit primes with a Digit length of 149

116^92 Contains all 21 consecutive 2 digit primes with a Digit length of 190

117^104 Contains all 21 consecutive 2 digit primes with a Digit length of 216

118^110 Contains all 21 consecutive 2 digit primes with a Digit length of 228

119^94 Contains all 21 consecutive 2 digit primes with a Digit length of 196

120^191 Contains all 21 consecutive 2 digit primes with a Digit length of 398

121^105 Contains all 21 consecutive 2 digit primes with a Digit length of 219

122^100 Contains all 21 consecutive 2 digit primes with a Digit length of 209

123^114 Contains all 21 consecutive 2 digit primes with a Digit length of 239

124^84 Contains all 21 consecutive 2 digit primes with a Digit length of 176

125^110 Contains all 21 consecutive 2 digit primes with a Digit length of 231

126^98 Contains all 21 consecutive 2 digit primes with a Digit length of 206

127^86 Contains all 21 consecutive 2 digit primes with a Digit length of 181

128^105 Contains all 21 consecutive 2 digit primes with a Digit length of 222

129^82 Contains all 21 consecutive 2 digit primes with a Digit length of 174

130^213 Contains all 21 consecutive 2 digit primes with a Digit length of 451

131^71 Contains all 21 consecutive 2 digit primes with a Digit length of 151

132^113 Contains all 21 consecutive 2 digit primes with a Digit length of 240

133^91 Contains all 21 consecutive 2 digit primes with a Digit length of 194

134^95 Contains all 21 consecutive 2 digit primes with a Digit length of 203

135^108 Contains all 21 consecutive 2 digit primes with a Digit length of 231

136^99 Contains all 21 consecutive 2 digit primes with a Digit length of 212

137^108 Contains all 21 consecutive 2 digit primes with a Digit length of 231

138^95 Contains all 21 consecutive 2 digit primes with a Digit length of 204

139^95 Contains all 21 consecutive 2 digit primes with a Digit length of 204

140^150 Contains all 21 consecutive 2 digit primes with a Digit length of 322

141^78 Contains all 21 consecutive 2 digit primes with a Digit length of 168

142^116 Contains all 21 consecutive 2 digit primes with a Digit length of 250

143^85 Contains all 21 consecutive 2 digit primes with a Digit length of 184

144^102 Contains all 21 consecutive 2 digit primes with a Digit length of 221

145^94 Contains all 21 consecutive 2 digit primes with a Digit length of 204

146^114 Contains all 21 consecutive 2 digit primes with a Digit length of 247

147^98 Contains all 21 consecutive 2 digit primes with a Digit length of 213

148^89 Contains all 21 consecutive 2 digit primes with a Digit length of 194

149^111 Contains all 21 consecutive 2 digit primes with a Digit length of 242

150^169 Contains all 21 consecutive 2 digit primes with a Digit length of 368

151^80 Contains all 21 consecutive 2 digit primes with a Digit length of 175

152^84 Contains all 21 consecutive 2 digit primes with a Digit length of 184

153^115 Contains all 21 consecutive 2 digit primes with a Digit length of 252

154^88 Contains all 21 consecutive 2 digit primes with a Digit length of 193

155^94 Contains all 21 consecutive 2 digit primes with a Digit length of 206

156^81 Contains all 21 consecutive 2 digit primes with a Digit length of 178

157^89 Contains all 21 consecutive 2 digit primes with a Digit length of 196

158^98 Contains all 21 consecutive 2 digit primes with a Digit length of 216

159^92 Contains all 21 consecutive 2 digit primes with a Digit length of 203

160^156 Contains all 21 consecutive 2 digit primes with a Digit length of 344

161^111 Contains all 21 consecutive 2 digit primes with a Digit length of 245

162^85 Contains all 21 consecutive 2 digit primes with a Digit length of 188

163^93 Contains all 21 consecutive 2 digit primes with a Digit length of 206

164^100 Contains all 21 consecutive 2 digit primes with a Digit length of 222

165^110 Contains all 21 consecutive 2 digit primes with a Digit length of 244

166^86 Contains all 21 consecutive 2 digit primes with a Digit length of 191

167^87 Contains all 21 consecutive 2 digit primes with a Digit length of 194

168^76 Contains all 21 consecutive 2 digit primes with a Digit length of 170

169^108 Contains all 21 consecutive 2 digit primes with a Digit length of 241

170^154 Contains all 21 consecutive 2 digit primes with a Digit length of 344

171^120 Contains all 21 consecutive 2 digit primes with a Digit length of 268

172^109 Contains all 21 consecutive 2 digit primes with a Digit length of 244

173^96 Contains all 21 consecutive 2 digit primes with a Digit length of 215

174^76 Contains all 21 consecutive 2 digit primes with a Digit length of 171

175^86 Contains all 21 consecutive 2 digit primes with a Digit length of 193

176^98 Contains all 21 consecutive 2 digit primes with a Digit length of 221

177^95 Contains all 21 consecutive 2 digit primes with a Digit length of 214

178^78 Contains all 21 consecutive 2 digit primes with a Digit length of 176

179^92 Contains all 21 consecutive 2 digit primes with a Digit length of 208

180^166 Contains all 21 consecutive 2 digit primes with a Digit length of 375

181^81 Contains all 21 consecutive 2 digit primes with a Digit length of 183

182^98 Contains all 21 consecutive 2 digit primes with a Digit length of 222

183^98 Contains all 21 consecutive 2 digit primes with a Digit length of 222

184^116 Contains all 21 consecutive 2 digit primes with a Digit length of 263

185^111 Contains all 21 consecutive 2 digit primes with a Digit length of 252

186^86 Contains all 21 consecutive 2 digit primes with a Digit length of 196

187^95 Contains all 21 consecutive 2 digit primes with a Digit length of 216

188^74 Contains all 21 consecutive 2 digit primes with a Digit length of 169

189^86 Contains all 21 consecutive 2 digit primes with a Digit length of 196

190^156 Contains all 21 consecutive 2 digit primes with a Digit length of 356

191^71 Contains all 21 consecutive 2 digit primes with a Digit length of 162

192^85 Contains all 21 consecutive 2 digit primes with a Digit length of 195

193^74 Contains all 21 consecutive 2 digit primes with a Digit length of 170

194^94 Contains all 21 consecutive 2 digit primes with a Digit length of 216

195^104 Contains all 21 consecutive 2 digit primes with a Digit length of 239

196^75 Contains all 21 consecutive 2 digit primes with a Digit length of 172

197^96 Contains all 21 consecutive 2 digit primes with a Digit length of 221

198^114 Contains all 21 consecutive 2 digit primes with a Digit length of 262

199^117 Contains all 21 consecutive 2 digit primes with a Digit length of 269.

Also, here is my "best smallest" digit length number containing all 2 digit primes.

47032626^8 = 23943841067583599349471291737363413331353119598979615802245376 Contains ->
{11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97} with a digit length of 62.

Other results:-

10 consecutive 3 digit primes.

2^1638 = 1222173597889602910459756565641663238761990577135273823494426661978265668250
92732887094283140056209570948058184113185092035960836796333041636233013553818118347
63465775225105626750353790297151369915699071270199571056859529425686869273749189843
32204540608756253653266473022500175206821694285995169538804535394427349011803415498
76692071831260933111929731896069474137390840398552613175617366589357181808240450946
83638257550135753955426279160789530239796571648665881414341172638502337744675687890
944 Contains -> {317,331,337,347,349,353,359,367,373,379} with a digit length of 494

10 consecutive 4 digit primes.

2^9971 = 3716094637067538471675928985110642544596450090196342123396711330586939832236980392
023913875974005944635594474044294864734924676148111704945116077723372042916315743708385861
390599095551532890628188950957156381255303045475568737481662167063889296655522303746993623
303110514140363382291340885585900872370672983159329117066122786469733645577999432174216613
489075708320297442905023835735810647735677934472254734758095715921262435793976535614778217
194105199358710193196224161077790006058178904640361042965072723716435167120056707790383695
475418047491134271726436919961662122467886811953891339646756263375545820951762746011491726
542893754154309220766081984280505054269362009658725094558358898250518436320659963978720481
179915671086995245995756503130186782151237362782684584186865387905316761927703580420509298
954309449704796105859520340058255166059889655846494986021353724657578596797867529711466320
405088543444213411958963767782145379492074785198367212325452082638893700374042783039548698
413549347120533710019895882533401221458222355551480884560344381799249954996321205419061222
015537764273741582973632720492743591442170272330683227733368042382506326172682048936056217
9989315341272891517462724704158368001877143749756106749863993074796429430647142741191336409
718007397632550095881535528699479950243831048913632113422634966481783485521439716145140648
130566227075240531282664631532593780909880667854073960562063463648004598671643658735981965
387531711177289683210292651788933976627412055428540813653339650877399271953125956421393418
266016964518089775630751509136793461855267303994445161096944848754318683999058439939093260
064433975768164304949683791035922109612011444901366720163920871563981238619796076480944005
413062794355649156767463641508705352857953793339027468266850304178629091734199278523970072
748952024760611208933691895916546908960832105935478636305931090532444659810490302473743154
739877971216334776663943242584622592942479102804736005688533362851006767254975439380905169
691434831184757684611233021116649670002156963123191334766875729296862941148469750624064408
806635827662339461365096996907138985100456165850896208311348783611574660758881001517597100
695800383516065211187398964782234695497802858730906910203802739970759233044940463791350520
423048755677030737478075923954404258669450208284956593621282945420495020684542966863304179
657154189437723011378879651964025638511630461250233634852652873565463293849702766983939319
100518998535976394371941743646282587144522418792792331470384937980352526783492742121591864
593857182655261995196775057096357963267874637647473223616596925617854810549852543399738078
232889927124658249992071674286086677145267087349017278397749188387258944845188650696236063
074164430266565725221724529886971334835366029637502221602608105926446715964076038606532703
644975855109542215244365583494671212329488200197593250067524367161538362910280227811528033
67249285459075951631836306584943996374919027383381152822589578057101942691614936255555008
012434969313560446140154286626305998848 Contains -> {6397,6421,6427,6449,6451,6469,6473,6481,6491,6521} with a digit length of 3002

b and c).

Solutions found with primes <=22801763489. (the 1000000000th prime).

Next Prime ->17*19 = 323 -> 23

Next Prime ->499*503 = 250997 -> 509

Next Prime ->1997*1999 = 3992003 -> 2003

Next Prime ->4999*5003 = 25009997 -> 5009

Next Prime ->9791*9803 = 95981173 -> 9811

Next Prime ->5999993*6000011 = 36000023999923 -> 6000023

Next Prime ->480083407*480083419 = 230480083437728533 -> 480083437

Next Prime ->600000019*600000041 = 360000036000000779 -> 600000077

** New one -> Next Prime ->6666533329*6666533333 = 44442666653333955557 -> 6666533339

Previous Prime ->5*7 = 35 -> 3

Previous Prime ->373*379 = 141367 -> 367

Previous Prime ->1999993*2000003 = 3999991999979 -> 1999979

Previous Prime ->37015607*37015621 = 1370155679796947 -> 37015567

** New one -> Previous Prime ->519794713*519794729 = 270186551979467777 -> 519794677

** New one -> Previous Prime ->2791000027*2791000073 = 7789681279100001971 -> 2791000019

** New one -> Previous Prime ->5236000027*5236000073 = 27415696523600001971 -> 5236000019

** New one -> Previous Prime ->5593000027*5593000073 = 31281649559300001971 -> 5593000019

** New one -> Previous Prime ->5656000027*5656000073 = 31990336565600001971 -> 5656000019

** New one -> Previous Prime ->6256000027*6256000073 = 39137536625600001971 -> 6256000019

** New one -> Previous Prime ->7170000023*7170000077 = 51408900717000001771 -> 7170000017

** New one -> Previous Prime ->7533000023*7533000077 = 56746089753300001771 -> 7533000017

** New one -> Previous Prime ->8996000023*8996000077 = 80928016899600001771 -> 8996000017

***

Metin wrote:

I get many results for p(n)^p(n+1) that contains three consecutive primes. So I checked for a further example...

An example for 115 consecutive primes following p(n) and p(n+1) of p(n)^p(n+1);

7621^7639 contains 115 consecutive primes following 7639 :{7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699}

++++++

Examples for p(n)^p(n+1) contains consecutive primes as concatenated: p(n+2)p(n+3)...

3 cons.: p(n)=37 {357} 37^41 contains 357 (conc. of 3,5,7)

4 cons. p(n)=43 {2357}

5 cons. p(n)=2423 {235711}

6 cons. p(n)=9109 {2,3,5,7,11,13}

7 cons. 194507^194521 contains 7 consecutive concatenated primes 2357111317 {conc. of 2,3,5,7,11,13,17}

Also 2^220811 contains {23571113} and 2^1748849 contains {2357111317}

b)

Next example I found is 6666533329 * 6666533333=44442666653333955557, contains the next prime: 6666533339.

c)

The product of two consecutive primes has the previous one in its representation;

Other ones I found are;

p(n) p(n+1) p(n-1)

{519794713, 519794729, 519794677} :519794713 * 519794729=270186551979467777

{2791000027, 2791000073, 2791000019}

{5236000027, 5236000073, 5236000019}

{5593000027, 5593000073, 5593000019}

{5656000027, 5656000073, 5656000019}

{6256000027, 6256000073, 6256000019}

***

Adam wrote:

I think it is a numbers game: if you get a lot of digits, you get a lot of different strings of digits and so a high probability of consecutive primes. For exponents between 1000 and 1200 I obtain the following 218 instances, including an exponent with 4 consecutive primes (exponent 1182, primes 5659,5669,5683,5689) as well as several 5 digits primes, including exponent 1175, primes 35543 & 35569 as well as primes 62653 & 62659. I limited my count to only primes over 1000. The data below is [exponent, first prime, next prime]:

[1001, 5347, 5351], [1002, 4297, 4327], [1002, 4561, 4567], [1003, 3833, 3847], [1003, 3847, 3851], [1004, 7187, 7193],
[1005, 9533, 9539], [1005, 9587, 9601], [1006, 2399, 2411], [1006, 7451, 7457], [1006, 9491, 9497], [1008, 6311, 6317],
[1009, 4561, 4567], [1011, 3593, 3607], [1013, 9769, 9781], [1014, 9551, 9587], [1015, 9661, 9677], [1016, 9323, 9337],
[1017, 1103, 1109], [1017, 4447, 4451], [1017, 761603, 761611], [1018, 7411, 7417], [1018, 9013, 9029], [1019, 3547, 3557],
[1023, 4057, 4073], [1023, 76781, 76801], [1026, 2531, 2539], [1026, 3083, 3089], [1027, 1579, 1583], [1028, 1559, 1567],
[1029, 9067, 9091], [1030, 8821, 8831], [1032, 2179, 2203], [1032, 6197, 6199], [1033, 3583, 3593], [1033, 6329, 6337],
[1033, 13001, 13003], [1034, 9871, 9883], [1035, 8167, 8171], [1036, 1471, 1481], [1036, 3919, 3923], [1036, 7687, 7691],
[1036, 8363, 8369], [1036, 9157, 9161], [1036, 9161, 9173], [1037, 8293, 8297], [1037, 8467, 8501], [1038, 29567, 29569],
[1039, 2657, 2659], [1039, 6899, 6907],
[1043, 9539, 9547], [1043, 9811, 9817], [1045, 8161, 8167], [1045, 8467, 8501], [1046, 1531, 1543], [1046, 1693, 1697],
[1046, 2017, 2027], [1046, 2161, 2179], [1046, 3049, 3061], [1046, 7549, 7559], [1046, 8543, 8563], [1048, 1109, 1117],
[1049, 2243, 2251], [1050, 1579, 1583], [1050, 3593, 3607], [1051, 1997, 1999], [1053, 4133, 4139], [1053, 89959, 89963],
[1055, 9007, 9011], [1056, 1033, 1039], [1059, 1429, 1433], [1060, 4567, 4583], [1060, 8089, 8093], [1060, 8219, 8221],
[1060, 8237, 8243], [1062, 5237, 5261], [1063, 2791, 2797], [1063, 7457, 7459], [1064, 7451, 7457], [1066, 7333, 7349],
[1066, 7481, 7487], [1068, 3709, 3719], [1069, 5209, 5227], [1071, 3463, 3467], [1071, 7499, 7507], [1072, 2383, 2389],
[1073, 1201, 1213], [1075, 1747, 1753], [1077, 5639, 5641], [1078, 1279, 1283], [1078, 2837, 2843], [1078, 89363, 89371],
[1081, 2309, 2311], [1081, 2311, 2333], [1081, 4651, 4657], [1084, 5879, 5881], [1086, 4129, 4133], [1086, 5189, 5197],
[1089, 3323, 3329], [1093, 6791, 6793], [1093, 14251, 14281], [1094, 2371, 2377], [1097, 2309, 2311], [1097, 7151, 7159],
[1099, 3373, 3389], [1102, 9397, 9403], [1103, 8461, 8467], [1104, 6917, 6947], [1105, 6361, 6367], [1106, 3181, 3187],
[1107, 6221, 6229], [1107, 6863, 6869], [1108, 3727, 3733], [1108, 3733, 3739], [1109, 5483, 5501], [1109, 5527, 5531],
[1110, 4651, 4657], [1111, 9479, 9491], [1111, 25793, 25799], [1112, 3583, 3593], [1113, 3001, 3011], [1114, 4363, 4373],
[1118, 4861, 4871], [1119, 1213, 1217], [1119, 7369, 7393], [1120, 3089, 3109], [1122, 1483, 1487], [1122, 4397, 4409],
[1123, 7459, 7477], [1124, 8819, 8821], [1125, 9629, 9631], [1129, 6211, 6217], [1129, 6217, 6221], [1133, 1163, 1171],
[1133, 2551, 2557], [1133, 5953, 5981], [1133, 6659, 6661], [1133, 7079, 7103], [1134, 6719, 6733], [1135, 6833, 6841],
[1136, 5273, 5279], [1137, 6577, 6581], [1139, 2843, 2851], [1140, 2621, 2633], [1141, 2113, 2129], [1141, 57059, 57073],
[1142, 3463, 3467], [1144, 2957, 2963], [1146, 3847, 3851], [1150, 4261, 4271], [1150, 4451, 4457], [1153, 9397, 9403],
[1153, 9403, 9413], [1155, 7589, 7591], [1156, 5527, 5531], [1156, 7001, 7013], [1156, 51829, 51839], [1159, 2143, 2153],
[1159, 4733, 4751], [1159, 7907, 7919], [1162, 5431, 5437], [1162, 7963, 7993], [1165, 5417, 5419], [1165, 9293, 9311],
[1166, 1721, 1723], [1166, 9787, 9791], [1168, 6553, 6563], [1168, 7927, 7933], [1168, 8867, 8887], [1169, 4783, 4787],
[1169, 6869, 6871], [1171, 4463, 4481], [1172, 2687, 2689], [1172, 7039, 7043], [1173, 7393, 7411], [1175, 3517, 3527],
[1175, 35543, 35569], [1175, 62653, 62659], [1177, 1789, 1801], [1177, 1889, 1901], [1178, 1181, 1187],
[1178, 1399, 1409], [1178, 9439, 9461], [1179, 4051, 4057], [1180, 7499, 7507], [1180, 9901, 9907], [1180, 25609, 25621],
[1182, 5471, 5477], [1182, 5659, 5669], [1182, 5669, 5683], [1182, 5683, 5689], [1184, 2609, 2617], [1184, 8179, 8191],
[1185, 8719, 8731], [1185, 8807, 8819], [1186, 2477, 2503], [1186, 3853, 3863], [1186, 6229, 6247], [1186, 7177, 7187],
[1187, 8191, 8209], [1189, 3739, 3761], [1189, 4513, 4517], [1189, 9397, 9403], [1189, 30697, 30703], [1190, 4093, 4099],
[1190, 5441, 5443], [1191, 8863, 8867], [1192, 2297, 2309], [1192, 15287, 15289], [1195, 9137, 9151], [1196, 5419, 5431],
[1196, 6173, 6197], [1197, 2837, 2843], [1197, 6571, 6577], [1198, 6781, 6791], [1199, 5309, 5323], [1199, 6311, 6317],
[1199, 8609, 8623]

***

Hakan wrote:

a) Here is a list in descending order of largest two consecutive primes in 2^exponent.

Largest two consecutive primes that I found is 7710473 and 7710481 in 2^2853 =
68956884131509655602858134732631917521878369001812510750557017399142893061947628586369121729237753974
79762743420166090149267266890162113214611677123575512718821841592527706987180785495773481462109453406
7710481377156575650112850681783360659787899091572781396803760888348826518206099013760701517174838388
04079388423713833434374398771268844767306552291329675985370451693872020208165373372426133880377639022
5119708667824182298771047359026520393591863622147887931753444048284002460426807569000925399996689097
963457332687012256860591888811292984097513714379767347669
86227906772124463807588631537154615836599975229172371242804379265739131777057247728709619794946561183
57670780278859156233356886704810580934179833747097760804529379254086991884034752246138670437174371468
1073034512350262535654412748686949218104570137508577522337125077104952695023669207897452673236992

Largest three consecutive primes that I found is 59243, 59263, 59273 in 2^3014 =
2015611981268774382684665959672631319987345646775228359911207778975840338927080159273549935146751524
17985273733783847716210750944198367121176179913861586670589535739962301647652949740767045331668602835
47480176728742862492724325171419760078966107183133381372457953901638852501314793087062896032415831601
1177348907446023113826949007459243887886721473704247925223896457396276823540036974733525472139752650
81913564923465792564898777203415594823430213979547115057621302468389904576037708791323570679766299832
759229877136541336925991930828366897524575367932848082215
10607686992776413691986339105323990459888437057528584963219478418905346118089218064111822604458778925
86716638419707612467922793513348505273701244466764529436314923895683843139542602337832482982075381171
4327118166692091773978428927175888356148306741452347799578223947905926304978493260779110621465623224
1546525266600139157895121772855594680145936384

b) 6666533329*6666533333 = 44442666653333955557 -> 6666533339

c) 519794713*519794729 = 270186551979467777 -> 519794677
2791000027*2791000073 = 7789681279100001971 -> 2791000019
5236000027*5236000073 = 27415696523600001971 -> 5236000019
5593000027*5593000073 = 31281649559300001971 -> 5593000019
5656000027*5656000073 = 31990336565600001971 -> 5656000019
6256000027*6256000073 = 39137536625600001971 -> 6256000019
7170000023*7170000077 = 51408900717000001771 -> 7170000017
7533000023*7533000077 = 56746089753300001771 -> 7533000017
8996000023*8996000077 = 80928016899600001771 -> 8996000017

See this file.

***

Simon wrote:

a)
There are so many solutions, I just picked a few examples:
3^108 = 9^54 contains first 10 primes.
11^2439 contains first 100 primes.
13^47596 contains first 1000 primes.
11^49669 contains first 1000 primes.
9^57111 contains first 1000 primes.
3^6636 contains 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601
11^109671 contains 224101, 224113, 224129, 224131, 224149, 224153, 224171, 224177, 224197

b) Searched up to 4*10^12:
17*19 = 323 contains 23
499*503 = 250997 contains 509
1997*1999 = 3992003 contains 2003
4999*5003 = 25009997 contains 5009
9791*9803 = 95981173 contains 9811
5999993*6000011 = 36000023999923 contains 6000023
480083407*480083419 = 230480083437728533 contains 480083437
600000019*600000041 = 360000036000000779 contains 600000077
6666533329*6666533333 = 44442666653333955557 contains 6666533339

c) Searched up to 4*10^12:
5*7 = 35 contains 3
373*379 = 141367 contains 367
1999993*2000003 = 3999991999979 contains 1999979
37015607*37015621 = 1370155679796947 contains 37015567
519794713*519794729 = 270186551979467777 contains 519794677
2791000027*2791000073 = 7789681279100001971 contains 2791000019
5236000027*5236000073 = 27415696523600001971 contains 5236000019
5593000027*5593000073 = 31281649559300001971 contains 5593000019
5656000027*5656000073 = 31990336565600001971 contains 5656000019
6256000027*6256000073 = 39137536625600001971 contains 6256000019
7170000023*7170000077 = 51408900717000001771 contains 7170000017
7533000023*7533000077 = 56746089753300001771 contains 7533000017
8996000023*8996000077 = 80928016899600001771 contains 8996000017
33960000023*33960000077 = 1153281603396000001771 contains 33960000017
44890000031*44890000069 = 2015112104489000002139 contains 44890000021
50840000023*50840000077 = 2584705605084000001771 contains 50840000017
77080000031*77080000069 = 5941326407708000002139 contains 77080000021
122310000023*122310000077 = 14959736112231000001771 contains 122310000017
128740000031*128740000069 = 16573987612874000002139 contains 128740000021
163040000023*163040000077 = 26582041616304000001771 contains 163040000017
167530000031*167530000069 = 28066300916753000002139 contains 167530000021
184400000023*184400000077 = 34003360018440000001771 contains 184400000017
188030000023*188030000077 = 35355280918803000001771 contains 188030000017
193049999981*193050000119 = 37268302519304999997739 contains 193049999977
207640000031*207640000069 = 43114369620764000002139 contains 207640000021
230030000023*230030000077 = 52913800923003000001771 contains 230030000017
246660000023*246660000077 = 60841155624666000001771 contains 246660000017
255220000027*255220000073 = 65137248425522000001971 contains 255220000019
260000000023*260000000077 = 67600000026000000001771 contains 260000000017
263980000027*263980000073 = 69685440426398000001971 contains 263980000019
279650000011*279650000089 = 78204122527965000000979 contains 279650000009
326770000031*326770000069 = 106778632932677000002139 contains 326770000021
327430000031*327430000069 = 107210404932743000002139 contains 327430000021
331240000027*331240000073 = 109719937633124000001971 contains 331240000019
338410000031*338410000069 = 114521328133841000002139 contains 338410000021
353650000031*353650000069 = 125068322535365000002139 contains 353650000021
379000000031*379000000069 = 143641000037900000002139 contains 379000000021
379300000031*379300000069 = 143868490037930000002139 contains 379300000021
400060000027*400060000073 = 160048003640006000001971 contains 400060000019
433240000031*433240000069 = 187696897643324000002139 contains 433240000021
438520000031*438520000069 = 192299790443852000002139 contains 438520000021
503800000027*503800000073 = 253814440050380000001971 contains 503800000019
506140000027*506140000073 = 256177699650614000001971 contains 506140000019
509840000023*509840000077 = 259936825650984000001771 contains 509840000017
519900000023*519900000077 = 270296010051990000001771 contains 519900000017
529020000023*529020000077 = 279862160452902000001771 contains 529020000017
541340000023*541340000077 = 293048995654134000001771 contains 541340000017
548620000031*548620000069 = 300983904454862000002139 contains 548620000021
568100000011*568100000089 = 322737610056810000000979 contains 568100000009
621190000027*621190000073 = 385877016162119000001971 contains 621190000019
631000000027*631000000073 = 398161000063100000001971 contains 631000000019
633520000027*633520000073 = 401347590463352000001971 contains 633520000019
635620000031*635620000069 = 404012784463562000002139 contains 635620000021
638840000023*638840000077 = 408116545663884000001771 contains 638840000017
702540000023*702540000077 = 493562451670254000001771 contains 702540000017
712420000031*712420000069 = 507542256471242000002139 contains 712420000021
759810000023*759810000077 = 577311236175981000001771 contains 759810000017
769390000031*769390000069 = 591960972176939000002139 contains 769390000021
777700000027*777700000073 = 604817290077770000001971 contains 777700000019
777850000027*777850000073 = 605050622577785000001971 contains 777850000019
791860000031*791860000069 = 627042259679186000002139 contains 791860000021
798230000023*798230000077 = 637171132979823000001771 contains 798230000017
832940000023*832940000077 = 693789043683294000001771 contains 832940000017
852850000031*852850000069 = 727353122585285000002139 contains 852850000021
894510000023*894510000077 = 800148140189451000001771 contains 894510000017
895110000023*895110000077 = 801221912189511000001771 contains 895110000017
908810000011*908810000089 = 825935616190881000000979 contains 908810000009
912920000023*912920000077 = 833422926491292000001771 contains 912920000017
956020000031*956020000069 = 913974240495602000002139 contains 956020000021
1166500000031*1166500000069 = 1360722250116650000002139 contains 1166500000021
1382800000027*1382800000073 = 1912135840138280000001971 contains 1382800000019
1507000000027*1507000000073 = 2271049000150700000001971 contains 1507000000019
1535900000023*1535900000077 = 2358988810153590000001771 contains 1535900000017
2146400000011*2146400000089 = 4607032960214640000000979 contains 2146400000009
2336000000023*2336000000077 = 5456896000233600000001771 contains 2336000000017
2793700000027*2793700000073 = 7804759690279370000001971 contains 2793700000019
2849500000031*2849500000069 = 8119650250284950000002139 contains 2849500000021
2856300000023*2856300000077 = 8158449690285630000001771 contains 2856300000017
3183400000031*3183400000069 = 10134035560318340000002139 contains 3183400000021
3209100000023*3209100000077 = 10298322810320910000001771 contains 3209100000017
3291800000011*3291800000089 = 10835947240329180000000979 contains 3291800000009
3364300000031*3364300000069 = 11318514490336430000002139 contains 3364300000021
3469000000031*3469000000069 = 12033961000346900000002139 contains 3469000000021
3529500000023*3529500000077 = 12457370250352950000001771 contains 3529500000017
3569200000027*3569200000073 = 12739188640356920000001971 contains 3569200000019
3834900000023*3834900000077 = 14706458010383490000001771 contains 3834900000017
3995500000031*3995500000069 = 15964020250399550000002139 contains 3995500000021

***

Oscar wrote:

Puzzle c.
Esaustive search up to 10^11 provides several more solutions.
p(n), p(n+1), p(n+2), p(n+1)*p(n+2)
519794677, 519794713, 519794729, 270186551979467777
2791000019, 2791000027, 2791000073, 7789681279100001971
5236000019, 5236000027, 5236000073, 27415696523600001971
5593000019, 5593000027, 5593000073, 31281649559300001971
5656000019, 5656000027, 5656000073, 31990336565600001971
6256000019, 6256000027, 6256000073, 39137536625600001971
7170000017, 7170000023, 7170000077, 51408900717000001771
7533000017, 7533000023, 7533000077, 56746089753300001771
8996000017, 8996000023, 8996000077, 80928016899600001771
33960000017, 33960000023, 33960000077, 1153281603396000001771
44890000021, 44890000031, 44890000069, 2015112104489000002139
50840000017, 50840000023, 50840000077, 2584705605084000001771
77080000021, 77080000031, 77080000069, 5941326407708000002139

An interesting pattern emerges. Most new solutions have the form:
p(n) = a*10^b+x
p(n+1) = a*10^b+y

p(n+2) = a*10^b+z

with constraints:

0 < x < y < z

y+z = 10^t
x = floor(y*z/(10^t))

a < 10^(b-t)

so that the representation of product p(n+1)*p(n+2) necessarily contains the representation of p(n), followed by t digits.

For t = 2, only four triplets (x,y,z) can generate many consecutive-prime solutions:

(21,31,69), (19,27,73), (17,23,77), (9,11,89).

As an example, I found a 200-digit solution for last triplet (9,11,89) by choosing a = 502379, b = 194:

p(n) = 502379*10^194 + 9

p(n+1) = 502379*10^194 + 11

p(n+2) = 502379*10^194 + 19

Puzzle b.
Exaustive search up to 10^11 provides only one more solution.
p(n), p(n+1), p(n+2), p(n)*p(n+1):
6666533329, 6666533333, 6666533339, 44442666653333955557

Three solutions have a general form as above, with a different set of constraints:

a = 5 or a = 6
t = b

x < 0 < y < z
z = a*(x+y)-1

I found only one more such solution, by choosing a = 6, b= 136, x = -47, y = 251, z = 1223

p(n) = 6*10^136 - 47

p(n+1) = 6*10^136 + 251

p(n+2) = 6*10^136 + 1223

Puzzle a.

Some champion powers 2^k with exponent k<1000:

2^780 contains primes 702199 and 702203

2^848 contains primes 9931, 9941, and 9949

2^962 contains all primes between 2 and 109

***

Emmanuel wrote:

First of all, I think that the smallest power of two that contains two consecutive primes is 32 = 2^5.
So, I made the list of the smallest a such that 2^a contains the n-th prime and the n+1-th prime :
n a
1 5
2 16
3 20
4 40
5 70
6 27
7 80
8 41
9 78
10 49
11 75
12 30
13 22
14 37
15 37
16 125
17 111
18 64
19 26
20 36
21 62
22 84
23 121
24 55
25 103
... (obtainable on demand)
166 121
167 348
168 724
(found in less than a second)
A big n mostly consumes more time. For instance, the primes 1299709, 1299721 (n = 100000) in 2^116256 are found in about 387 seconds.

It has been proven that there is a solution for every n.

I think the proof will be found in :

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1 pp. 29; 199-200 Prob. 91a Dover NY 1987.

That book is not in my personal library, so I have not yet seen that proof.

I found that reference in  A018856 (where 2^a(n) is the smallest power of 2 that begins with n ). I was lead to that topic by an article in a Flemish journal.

Since there exists a power of two that begins with any number you want, you can take the power of two that begins with the concatenation of the nth and the (n+1)-th prime

(or eventually by the concatenation of the first n primes). Of course, that power of two in general will not be the smallest.

There can be many consecutive primes in powers of two,for instance :

Smallest a such that 2^a contains the first n primes
n a
1 1
2 5
3 25
4 29
5 58
6 70
7 70
8 162
9 162
10 269
11 321
12 330
13 345
14 364
15 400
16 400
17 400
18 513
19 513
20 513
21 513
22 563
23 563
24 563
25 563

For other bases than 2 I restricted myself to :

Smallest a such that a^2 contains the n-th and the n+1-th prime
n a
1 18
2 55
3 24
5 343
6 337
7 363
8 1781
9 486
10 427
11 177
12 1854
13 371
14 379
15 1573
16 1074
17 244
18 769
19 819
20 1293
21 1318
22 1214
23 3852
24 1133
25 824
... (obtainable on demand and easy to get)
168 54753

Smallest a such that a^2 contains the first n primes :
n a (its square is in A263404 at the OEIS.
1 5
2 18
3 55
4 189
5 3425
6 8434
7 33654
8 130905
9 1386835
10 7628323
11 7628323
12 15210923
13 41501523
14 1064965877
( It took a long time to get there : the next ones are copied from A263404)417
15 21740371527
16 131522291629
17 647860873427
18 2078276055077

Smallest a such that a^3 contains the first n primes.
n a
1 3
2 18
3 18
4 59
5 417
6 417
7 417
8 24724
9 121017
10 1212057
11 1244655
12 1244655
13 5243648
14 106937895
15 150802775
16 1096944544
17 4904053211
See the complete results obtained by Emmanuel in the following files 1 & 2.

***

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