Problems & Puzzles: Puzzles

Puzzle 1013. Primes as subsequences in pi

Carlos Rivera generated the sequence of primes P-champions in the quantity of digits needed to find P as a subsequence in π. Here is the sequence for all the primes below 2^32

 Prime champion Total of digits in π Position for each digit 2 7 7 7 14 14 61 38 8 38 103 44 2 33 44 1009 56 2 33 51 56 2003 65 7 33 51 65 2141 69 7 38 58 69 10039 80 2 33 51 65 80 10061 95 2 33 51 70 95 10067 97 2 33 51 70 97 100613 112 2 33 51 70 95 112 1001159 123 2 33 51 69 95 110 123 2001731 139 7 33 51 69 97 112 139 10011359 145 2 33 51 69 95 112 131 145 10017311 149 2 33 51 69 97 112 139 149 100113199 170 2 33 51 69 95 112 139 145 170 100155563 195 2 33 51 69 91 110 131 182 195 100175561 199 2 33 51 69 97 110 131 182 199 100175567 210 2 33 51 69 97 110 131 182 210 1001131613 216 2 33 51 69 95 112 139 182 199 216 1001735671 221 2 33 51 69 97 112 131 182 210 221 1001735677 225 2 33 51 69 97 112 131 182 210 225

Example:

The prime 1001735677 is the first prime P that needs 225 digits from π to get P as a subsequence:

314159265358979323846264338327950288419716939937510582097494459230781
640628620899862803482534211
706798214808651328230664709384460955058223
1725359408128481117450284102701938521105559
64462294895493038196442881

097566593344612847 (225 digits)

Q1. Can you extend this table?

Q2. Construct the corresponding table of Prime-Champions in the digits for

2 for all the primes below 2^32

Q3. What should be the common features for both tables, if any?

During the week 16-21, August, 2020, contributions came from Jeff Heleen, Emmanuel Vantieghem

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Jeff wrote:

The limit of my search was 1.5*10^9, but I did it for sqrt(2), sqrt(3) and e:

 sqrt(2) prime total digits of sqrt(2) Position for each digit 2 5 5 3 7 7 5 8 8 7 12 12 19 15 1, 15 31 22 7, 22 83 47 19, 47 839 50 19, 47, 50 1031 56 1, 14, 47, 56 8311 77 19, 47, 56, 77 8329 113 19, 47, 70, 113 83299 117 19, 47, 70, 113, 117 83597 118 19, 47, 84, 113, 118 89293 126 19, 31, 70, 113, 126 832933 135 19, 47, 70, 113, 126, 135 832969 147 19, 47, 70, 113, 127, 147 8329213 153 19, 47, 70, 113, 115, 138, 153 8329691 157 19, 47, 70, 113, 127, 147, 157 83290139 167 19, 47, 70, 113, 119, 138, 153, 167 83291137 169 19, 47, 70, 113, 114, 138, 153, 169 83293921 187 19, 47, 70, 113, 126, 147, 161, 187 83295923 222 19, 47, 70, 113, 129, 147, 161, 222 832939631 231 19, 47, 70, 113, 126, 147, 164, 222, 231 832959637 233 19, 47, 70, 113, 129, 147, 164, 222, 233 832964633 257 19, 47, 70, 113, 127, 141, 164, 222, 257
 sqrt(3) prime total digits of sqrt(3) Position for each digit 2 4 4 5 6 6 11 28 1, 28 47 33 23, 33 113 35 1, 28, 35 149 38 1, 23, 38 211 48 4, 28, 38 499 62 23, 38, 62 997 63 18, 38, 63 2111 67 4, 28, 48, 67 4273 69 23, 34, 63, 69 11117 82 1, 28, 48, 67, 82 42727 99 23, 34, 63, 95, 99 42773 118 23, 34, 63, 82, 118 47779 121 23, 33, 63, 82, 121 427237 138 23, 34, 63, 95, 118, 138 427733 152 23, 34, 63, 82, 118, 152 4272377 153 23, 34, 63, 95, 118, 138, 153 4278931 166 23, 34, 63, 76, 121, 152, 166 42723319 167 23, 34, 63, 95, 118, 152, 166, 167 42723367 178 23, 34, 63, 95, 118, 152, 155, 178 42729311 185 23, 34, 63, 95, 121, 152, 166, 185 427231873 196 23, 34, 63, 95, 118, 141, 157, 178, 196
 e prime total digits of e Position for each digit 2 1 1 3 18 18 17 25 3, 25 31 28 18, 28 109 36 3, 14, 36 311 85 18, 28, 85 1783 107 3, 25, 87, 107 1789 108 3, 25, 87, 108 17807 124 3, 25, 87, 113, 124 178067 138 3, 25, 87, 113, 130, 138 178361 161 3, 25, 87, 107, 130, 161 1780483 162 3, 25, 87, 113, 125, 160, 162 1780487 169 3, 25, 87, 113, 125, 160, 169 1780489 170 3, 25, 87, 113, 125, 160, 170 3183611 191 18, 28, 87, 107, 130, 161, 191 17806111 196 3, 25, 87, 113, 130, 161, 191, 196 17834857 204 3, 25, 87, 107, 125, 160, 189, 204 31806113 205 18, 28, 87, 113, 130, 161, 191, 205 178048043 213 3, 25, 87, 113, 125, 160, 175, 208, 213 178048567 235 3, 25, 87, 113, 125, 160, 189, 234, 235 178061563 271 3, 25, 87, 113, 130, 161, 189, 234, 271

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Emmanuel wrote:

Q1.
The list continues with
10011316729     248      2,33,51,69,95,112,139,182,210,222,248
10061556707     289      2,33,51,70,95,110,131,182,210,246,289

Q2
The list looks like this :
2      5      5
3      7      7
5      8      8
7      12      12
19      15      1, 15
31      22      7, 22
83      47      19, 47
839      50      19, 47, 50
1031      56      1, 14, 47, 56
8311      77      19, 47, 56, 77
8329      113      19, 47, 70, 113
83299      117      19, 47, 70, 113, 117
83597      118      19, 47, 84, 113, 118
89293      126      19, 31, 70, 113, 126
832933      135      19, 47, 70, 113, 126, 135
832969      147      19, 47, 70, 113, 127, 147
8329213      153      19, 47, 70, 113, 115, 138, 153
8329691      157      19, 47, 70, 113, 127, 147, 157
83290139      167      19, 47, 70, 113, 119, 138, 153, 167
83291137      169      19, 47, 70, 113, 114, 138, 153, 169
83293921      187      19, 47, 70, 113, 126, 147, 161, 187
83295923      222      19, 47, 70, 113, 129, 147, 161, 222
832939631      231      19, 47, 70, 113, 126, 147, 164, 222, 231
832959637      233      19, 47, 70, 113, 129, 147, 164, 222, 233
832964633      257      19, 47, 70, 113, 127, 141, 164, 222, 257

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I ended the computation when  p  was 1326341171 because there was not enough time.

But I'm almost sure the next  prime wil be > 8329000000.

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