Following on from my previous post, let's consider primes such that, when we add their consecutive pairs of digits together and concatenate them, we end up with another prime. Let's consider an example. I recently turned \( \textbf{27901}\) days old. This number is prime and this is what happens when we apply these operations to the number:$$ \underbrace{2+7}_{ \Large 9} \quad \underbrace{7+9}_{\Large 16} \quad \underbrace{9+0}_{ \Large 9} \quad \underbrace{0+1}_{ \Large 1} \rightarrow 91691 \text{ a prime number} $$In the range up to 40000, there are 403 numbers that satisfy the criteria. I'll show the 101 numbers from 27901 to 40000 below (permalink):
27901, 28081, 28163, 28201, 28229, 28447, 28607, 28621, 28627, 28661, 28703, 28867, 28927, 28961, 29009, 29021, 29063, 29129, 29167, 29201, 29207, 29221, 29401, 29567, 29789, 29881, 29921, 30029, 30089, 30109, 30427, 30509, 30643, 30649, 30763, 30781, 30809, 30829, 30983, 31249, 31627, 31663, 31849, 31883, 32009, 32089, 32189, 32401, 32429, 32563, 32609, 32749, 32843, 32909, 33029, 33149, 33289, 33349, 33529, 33601, 33829, 33889, 34147, 34261, 34327, 34483, 34501, 34583, 34589, 34807, 34963, 35089, 35129, 35401, 35461, 35543, 35603, 36109, 36209, 36229, 36721, 36901, 37021, 37189, 37409, 37589, 37747, 37963, 38047, 38149, 38461, 38629, 38707, 38729, 39103, 39229, 39409, 39503, 39749, 39829, 39929
Let's take the last number, 39929, as a second example:$$ \underbrace{3+9}_{ \Large 12} \quad \underbrace{9+9}_{\Large 18} \quad \underbrace{9+2}_{ \Large 11} \quad \underbrace{2+9}_{ \Large 11} \rightarrow 12181111 \text{ a prime number} $$For completeness, here are primes that arise from the above primes:
27901 --> 9169128081 --> 1088928163 --> 1097928201 --> 10102128229 --> 101041128447 --> 101281128607 --> 10146728621 --> 10148328627 --> 10148928661 --> 101412728703 --> 10157328867 --> 1016141328927 --> 101711928961 --> 101715729009 --> 1190929021 --> 1192329063 --> 1196929129 --> 111031129167 --> 111071329201 --> 11112129207 --> 11112729221 --> 11114329401 --> 11134129567 --> 1114111329789 --> 1116151729881 --> 111716929921 --> 111811330029 --> 3021130089 --> 3081730109 --> 311930427 --> 346930509 --> 355930643 --> 3610730649 --> 36101330763 --> 3713930781 --> 3715930809 --> 388930829 --> 38101130983 --> 39171131249 --> 4361331627 --> 478931663 --> 4712931849 --> 49121331883 --> 49161132009 --> 520932089 --> 5281732189 --> 5391732401 --> 564132429 --> 5661132563 --> 5711932609 --> 586932749 --> 59111332843 --> 51012732909 --> 5119933029 --> 6321133149 --> 6451333289 --> 65101733349 --> 6671333529 --> 6871133601 --> 696133829 --> 611101133889 --> 611161734147 --> 7551134261 --> 768734327 --> 775934483 --> 78121134501 --> 795134583 --> 79131134589 --> 79131734807 --> 7128734963 --> 71315935089 --> 8581735129 --> 8631135401 --> 894135461 --> 8910735543 --> 8109735603 --> 8116336109 --> 971936209 --> 982936229 --> 9841136721 --> 9139336901 --> 9159137021 --> 1072337189 --> 10891737409 --> 10114937589 --> 1012131737747 --> 1014111137963 --> 101615938047 --> 11841138149 --> 11951338461 --> 111210738629 --> 111481138707 --> 11157738729 --> 111591139103 --> 12101339229 --> 121141139409 --> 12134939503 --> 12145339749 --> 1216111339829 --> 1217101139929 --> 12181111
What if we convert the prime numbers in base 10 to numbers in other bases. These numbers will remain prime of course and then we apply the operations but in our chosen base. Let's work with base 9. In the range from 27901 up to 40000, 121 numbers satisfy (permalink):
28123, 28211, 28279, 28297, 28463, 28513, 28517, 28549, 28621, 28663, 28697, 28711, 28789, 28859, 28933, 29167, 29221, 29251, 29269, 29399, 29527, 29581, 29669, 29683, 29759, 29761, 29917, 29921, 30013, 30133, 30139, 30187, 30553, 30677, 30763, 30839, 30841, 30871, 30949, 31159, 31181, 31219, 31249, 31271, 31357, 31397, 31489, 31541, 32059, 32077, 32083, 32099, 32353, 32491, 32497, 32621, 32653, 32783, 32917, 32939, 32987, 33149, 33331, 33461, 33479, 33581, 33617, 33619, 33641, 33851, 34019, 34033, 34127, 34211, 34303, 34429, 34537, 34607, 34757, 34897, 34913, 34919, 35081, 35291, 35363, 35401, 35423, 35509, 35617, 35671, 35747, 35837, 35869, 35923, 35963, 35993, 36011, 36389, 36479, 36571, 36767, 36857, 36929, 37199, 37273, 37307, 37313, 37361, 37397, 37997, 38083, 38261, 38321, 38447, 38461, 38677, 38933, 39041, 39233, 39239, 39857
Let's look at 39239, the second last number in that list. It becomes 58738 in base 9.$$ \begin{align} 28219_{\, 10} = 58738_{ \, 9} &\rightarrow \underbrace{5+8}_{ \Large 14} \quad \underbrace{8+7}_{\Large 16} \quad \underbrace{7+3}_{ \Large 11} \quad \underbrace{3+8}_{ \Large 12}\\ \\ &\rightarrow 14161112_{\,9} \text{ a prime number} \\ \\ &= 7007969_{\,10} \text{ also prime} \end{align}$$The details are as follows (first column is base 10 prime, second column is prime in base 9, third prime is the prime resulting from the operations of addition and concatenation in base 9 and the fourth column is the prime converted to a prime in base 10.
28123 --> 42517 --> 6768 --> 5003 28211 --> 42625 --> 6887 --> 5101 28279 --> 42711 --> 61082 --> 40169 28297 --> 42731 --> 610114 --> 360949 28463 --> 43035 --> 7338 --> 5381 28513 --> 43101 --> 7411 --> 5437 28517 --> 43105 --> 7415 --> 5441 28549 --> 43141 --> 7455 --> 5477 28621 --> 43231 --> 7554 --> 5557 28663 --> 43277 --> 751015 --> 446891 28697 --> 43325 --> 7657 --> 5641 28711 --> 43341 --> 7675 --> 5657 28789 --> 43437 --> 77711 --> 51607 28859 --> 43525 --> 7877 --> 5821 28933 --> 43617 --> 71078 --> 46727 29167 --> 44007 --> 8407 --> 6163 29221 --> 44067 --> 84614 --> 55903 29251 --> 44111 --> 8522 --> 6257 29269 --> 44131 --> 8544 --> 6277 29399 --> 44285 --> 861114 --> 512581 29527 --> 44447 --> 88812 --> 58979 29581 --> 44517 --> 81068 --> 53279 29669 --> 44625 --> 81187 --> 53377 29683 --> 44641 --> 811115 --> 479777 29759 --> 44735 --> 812118 --> 480509 29761 --> 44737 --> 8121111 --> 4324519 29917 --> 45031 --> 10534 --> 6997 29921 --> 45035 --> 10538 --> 7001 30013 --> 45147 --> 106512 --> 63839 30133 --> 45301 --> 10831 --> 7237 30139 --> 45307 --> 10837 --> 7243 30187 --> 45361 --> 108107 --> 64969 30553 --> 45817 --> 1014108 --> 541007 30677 --> 46065 --> 116612 --> 70481 30763 --> 46171 --> 11788 --> 7937 30839 --> 46265 --> 118812 --> 72101 30841 --> 46267 --> 118814 --> 72103 30871 --> 46311 --> 111042 --> 66377 30949 --> 46407 --> 111147 --> 66463 31159 --> 46661 --> 1113137 --> 599353 31181 --> 46685 --> 11131514 --> 5394289 31219 --> 46737 --> 11141111 --> 5400523 31249 --> 46771 --> 1114158 --> 600101 31271 --> 46805 --> 111585 --> 66821 31357 --> 47011 --> 12712 --> 8597 31397 --> 47055 --> 127511 --> 77689 31489 --> 47167 --> 128714 --> 78583 31541 --> 47235 --> 121058 --> 72953 32059 --> 47871 --> 1216168 --> 660617 32077 --> 48001 --> 13801 --> 9397 32083 --> 48007 --> 13807 --> 9403 32099 --> 48025 --> 13827 --> 9421 32353 --> 48337 --> 1312611 --> 717103 32491 --> 48511 --> 131462 --> 79841 32497 --> 48517 --> 131468 --> 79847 32621 --> 48665 --> 13151312 --> 6470129 32653 --> 48711 --> 131682 --> 80021 32783 --> 48865 --> 13171512 --> 6483413 32917 --> 50134 --> 5147 --> 3769 32939 --> 50158 --> 51614 --> 34033 32987 --> 50222 --> 5244 --> 3847 33149 --> 50422 --> 5464 --> 4027 33331 --> 50644 --> 56118 --> 37277 33461 --> 50808 --> 5888 --> 4373 33479 --> 50828 --> 581111 --> 348553 33581 --> 51052 --> 6157 --> 4507 33617 --> 51102 --> 6212 --> 4547 33619 --> 51104 --> 6214 --> 4549 33641 --> 51128 --> 62311 --> 41077 33851 --> 51382 --> 641211 --> 381439 34019 --> 51588 --> 661417 --> 394729 34033 --> 51614 --> 6775 --> 5009 34127 --> 51728 --> 681011 --> 407521 34211 --> 51832 --> 610125 --> 360959 34303 --> 52044 --> 7248 --> 5309 34429 --> 52204 --> 7424 --> 5449 34537 --> 52334 --> 7567 --> 5569 34607 --> 52422 --> 7664 --> 5647 34757 --> 52608 --> 7868 --> 5813 34897 --> 52774 --> 7101512 --> 3780281 34913 --> 52802 --> 71182 --> 46811 34919 --> 52808 --> 71188 --> 46817 35081 --> 53108 --> 8418 --> 6173 35291 --> 53362 --> 86108 --> 56951 35363 --> 53452 --> 87107 --> 57679 35401 --> 53504 --> 8854 --> 6529 35423 --> 53528 --> 88711 --> 58897 35509 --> 53634 --> 810107 --> 479041 35617 --> 53764 --> 8111411 --> 4318201 35671 --> 53834 --> 812127 --> 480517 35747 --> 54028 --> 104211 --> 62137 35837 --> 54138 --> 105412 --> 63029 35869 --> 54174 --> 105812 --> 63353 35923 --> 54244 --> 10668 --> 7109 35963 --> 54288 --> 1061117 --> 571633 35993 --> 54332 --> 10765 --> 7187 36011 --> 54352 --> 10787 --> 7207 36389 --> 54822 --> 1013114 --> 540283 36479 --> 55032 --> 11535 --> 7727 36571 --> 55144 --> 11658 --> 7829 36767 --> 55382 --> 1181211 --> 643879 36857 --> 55502 --> 111152 --> 66467 36929 --> 55582 --> 11111411 --> 5381083 37199 --> 56022 --> 12624 --> 8527 37273 --> 56114 --> 12725 --> 8609 37307 --> 56152 --> 12767 --> 8647 37313 --> 56158 --> 127614 --> 77773 37361 --> 56222 --> 12844 --> 8707 37397 --> 56262 --> 12888 --> 8747 37997 --> 57108 --> 13818 --> 9413 38083 --> 57214 --> 131035 --> 79493 38261 --> 57432 --> 131275 --> 79691 38321 --> 57508 --> 131358 --> 79757 38447 --> 57658 --> 13141214 --> 6463489 38461 --> 57674 --> 13141412 --> 6463649 38677 --> 58044 --> 14848 --> 10169 38933 --> 58358 --> 1412814 --> 776317 39041 --> 58488 --> 14131317 --> 6988453 39233 --> 58732 --> 1416115 --> 778667 39239 --> 58738 --> 14161112 --> 7007969 39857 --> 60605 --> 6665 --> 4919
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