Talk:Rare numbers: Difference between revisions
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::I see this BigInteger conversion attempt as a means of reaching a few 21 digit numbers (which it did) and a means to reveal any shortcomings in the existing ulong algorithm which don't translate well to the BigInteger version (which it also did, I guess). If only it wasn't so impractically slow. If I could get it an order of magnitude faster, it would easier to persue this further. Creating a multi-tasking version would probably only go 2 to 4 times faster.--[[User:Enter your username|Enter your username]] ([[User talk:Enter your username|talk]]) 06:57, 24 October 2019 (UTC)
There only seem to be 5 21 digit rare numbers, so I started looking at 22 digits. I found that each odd number of digits takes less time (about 80% of the time) of the previous even number of digits, but each even/odd pair (number of digits such as 16/17 vs 14/15) takes about 20 times a long as the previous pair, so the computation time increases dramatically for results above 19 digits. Here are
<font color=blue>'''S.No.''' '''R''' '''R+R1''' '''R-R1'''</font>
85 208393425242000083802 20415029402<sup>2</sup> 115866300<sup>2</sup>
86 219518549668074815912 20953210268<sup>2</sup> 8877000<sup>2</sup>
87 257661195832219326752 22699892248<sup>2</sup> 193089600<sup>2</sup>
88 286694688797362186682 23945269942<sup>2</sup> 115866300<sup>2</sup>
89 837982875780054779738 40938494426<sup>2</sup> 73659300<sup>2</sup>
90 2414924301133245383042 69417286928<sup>2</sup> 3329996670<sup>2</sup>
91 2414924323311045383042 69417286928<sup>2</sup> 3330003330<sup>2</sup>
92 2414946523311023183042 69417286928<sup>2</sup> 3336663330<sup>2</sup>
93 2576494891793995836752 71783569748<sup>2</sup> 329996700<sup>2</sup>
94 2576494893971995836752 1783569748<sup>2</sup> 330003300<sup>2</sup>
95 2620937863931054483162 72351795868<sup>2</sup> 2663336730<sup>2</sup>
96 2620955623931476283162 72351795868<sup>2</sup> 2669996730<sup>2</sup>
97 2620955641393276283162 72351795868<sup>2</sup> 2670003270<sup>2</sup>
98 2622935621573476481162 72351795868<sup>2</sup> 3329996670<sup>2</sup>
99 2622935643751276481162 72351795868<sup>2</sup> 3330003330<sup>2</sup>
100 2622937641933274481162 72351795868<sup>2</sup> 3330603330<sup>2</sup>
101 2622955841933256281162 72351795868<sup>2</sup> 3336063330<sup>2</sup>
102 2622957843751254281162 72351795868<sup>2</sup> 3336663330<sup>2</sup>
103 2727651947516658327272 73857230612<sup>2</sup> 642947400<sup>2</sup>
104 2747736918335953517072 73857230612<sup>2</sup> 6370504140<sup>2</sup>
105 2788047668617596408872 74673252412<sup>2</sup> 26673000<sup>2</sup>
106 2788047848617776408872 74673252412<sup>2</sup> 32733000<sup>2</sup>
107 2788047868437576408872 74673252412<sup>2</sup> 33333000<sup>2</sup>
108 2788047888617376408872 74673252412<sup>2</sup> 33933000<sup>2</sup>
109 2939501759705522349392 76674206972<sup>2</sup> 263637300<sup>2</sup>
110 2939503375709360349392 76674206972<sup>2</sup> 269697300<sup>2</sup>
111 2939503537707740349392 76674206972<sup>2</sup> 270297300<sup>2</sup>
112 2939521359525562149392 76674206972<sup>2</sup> 329703300<sup>2</sup>
113 2939521557527542149392 76674206972<sup>2</sup> 330303300<sup>2</sup>
114 2939523577527340149392 76674206972<sup>2</sup> 336363300<sup>2</sup>
115 2939523779525320149392 76674206972<sup>2</sup> 336963300<sup>2</sup>
116 2959503377707360349192 76674206972<sup>2</sup> 6330303360<sup>2</sup>
<font color=red> '''117''' '''6344828989519887483525''' '''107697153531'''<sup>'''2'''</sup> '''33030003033'''<sup>'''2'''</sup></font>
118 8045841652464561594308 126810067846<sup>2</sup> 3299999670<sup>2</sup>
119 8045841654642561594308 126810067846<sup>2</sup> 3300000330<sup>2</sup>
120 8655059576513659814468 131526610006<sup>2</sup> 3296970330<sup>2</sup>
121 8655059772157639814468 131526610006<sup>2</sup> 3297029670<sup>2</sup>
122 8655079374155679614468 131526610006<sup>2</sup> 3302969670<sup>2</sup>
123 8655079574515659614468 131526610006<sup>2</sup> 3303030330<sup>2</sup>
That took about 5 3/4 days of computation. Only one odd number. Lots of 2,2 combinations there. Nearly half of all rare numbers under 21 digits start and end with 2. The trend continues above 20 digits with over half starting and ending with 2. --[[User:Enter your username|Enter your username]] ([[User talk:Enter your username|talk]]) 06:17, 2 December 2019 (UTC)
:Now that you've established that there are five 21 digit rare numbers, it might be worth mailing SSG (at guptass@rediffmail.com) to see if he'll add them to his list.
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