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improved value for N=10^40
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Oleg567
Oleg567
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10^n  + delta             =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391          =        2150209 ^2 * 2162911        26.4833
10^22 + 782939691         =       15634603 ^2 * 40909699       29.5443
10^25 + 44473963077       =      164317583 ^2 * 370366693      35.3722
10^28 + 192056856217      =     1417504619 ^2 * 4976809697     37.4827
10^31 + 1903633770933     =    13116605573 ^2 * 58124212477    40.7919        
10^34 + 4064603456403     =   245902790183 ^2 * 165376229827   41.8863
10^37 + 1483238923930317  =  1102689419521 ^2 * 8224198521037  50.3977
10^40 + 1968920864061936111233721191891321 = 1711540759862310229370048307 ^2 * 3413701221980995565738654929 5453.12833187 (???)

10^n  + delta             =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391          =        2150209 ^2 * 2162911        26.4833
10^22 + 782939691         =       15634603 ^2 * 40909699       29.5443
10^25 + 44473963077       =      164317583 ^2 * 370366693      35.3722
10^28 + 192056856217      =     1417504619 ^2 * 4976809697     37.4827
10^31 + 1903633770933     =    13116605573 ^2 * 58124212477    40.7919        
10^34 + 4064603456403     =   245902790183 ^2 * 165376229827   41.8863
10^37 + 1483238923930317  =  1102689419521 ^2 * 8224198521037  50.3977
10^40 + 19689208640619361 = 17115407598623 ^2 * 34137012219809 54.1283 (???)

10^n  + delta             =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391          =        2150209 ^2 * 2162911        26.4833
10^22 + 782939691         =       15634603 ^2 * 40909699       29.5443
10^25 + 44473963077       =      164317583 ^2 * 370366693      35.3722
10^28 + 192056856217      =     1417504619 ^2 * 4976809697     37.4827
10^31 + 1903633770933     =    13116605573 ^2 * 58124212477    40.7919        
10^34 + 4064603456403     =   245902790183 ^2 * 165376229827   41.8863
10^37 + 1483238923930317  =  1102689419521 ^2 * 8224198521037  50.3977
10^40 + 11233721191891321 = 10229370048307 ^2 * 95565738654929 53.3187 (???)
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Oleg567
Oleg567
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10^n  + delta             =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391          =        2150209 ^2 * 2162911        26.4833
10^22 + 782939691         =       15634603 ^2 * 40909699       29.5443
10^25 + 44473963077       =      164317583 ^2 * 370366693      35.3722
10^28 + 192056856217      =     1417504619 ^2 * 4976809697     37.4827
10^31 + 1903633770933     =    13116605573 ^2 * 58124212477    40.7919        
10^34 + 4064603456403     =   245902790183 ^2 * 165376229827   41.8863
10^37 + 1483238923930317  =  1102689419521 ^2 * 8224198521037  50.3977
10^40 + 19689208640619361 = 17115407598623 ^2 * 34137012219809 54.1283 (???)

10^n  + delta            =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391         =       2150209 ^2 * 2162911       26.4833
10^22 + 782939691        =      15634603 ^2 * 40909699      29.5443
10^25 + 44473963077      =     164317583 ^2 * 370366693     35.3722
10^28 + 192056856217     =    1417504619 ^2 * 4976809697    37.4827
10^31 + 1903633770933    =   13116605573 ^2 * 58124212477   40.7919        
10^34 + 4064603456403    =  245902790183 ^2 * 165376229827  41.8863
10^37 + 1483238923930317 = 1102689419521 ^2 * 8224198521037 50.3977

10^n  + delta             =              p ^2 * q          log_2(delta)
---------------------------------------------------------------------
10^19 + 93815391          =        2150209 ^2 * 2162911        26.4833
10^22 + 782939691         =       15634603 ^2 * 40909699       29.5443
10^25 + 44473963077       =      164317583 ^2 * 370366693      35.3722
10^28 + 192056856217      =     1417504619 ^2 * 4976809697     37.4827
10^31 + 1903633770933     =    13116605573 ^2 * 58124212477    40.7919        
10^34 + 4064603456403     =   245902790183 ^2 * 165376229827   41.8863
10^37 + 1483238923930317  =  1102689419521 ^2 * 8224198521037  50.3977
10^40 + 19689208640619361 = 17115407598623 ^2 * 34137012219809 54.1283 (???)
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Oleg567
Oleg567
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The smallest "diamond-numbers" (again, if mono mistakes) related to numbers of the form $10^{3k+1}$, are:

The smallest "diamond-numbers" (again, if mo mistakes) related to numbers of the form $10^{3k+1}$, are:

The smallest "diamond-numbers" (again, if no mistakes) related to numbers of the form $10^{3k+1}$, are:

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Oleg567
Oleg567
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Source Link
Oleg567
Oleg567
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