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# Some “Incredible” Factorizations of Special Integers

## Table A

This table contains the prime factorization of large integers defined by an algebraic formula. The integers are selected with two criteria:

1. They have no obvious factorization.
2. Current general factorization methods are either difficult or impossible to apply.

The table contains following information:

• Only prime factors with 41 or more digits are shown.
• In brackets is the number of digits and the outcome of the primality test.
• The critical factor (i.e. the 2nd largest prime factor known), which usually determines the difficulty of factorization, is highlighted in blue.
• Composite factors are shown in red.
 Integer y Digits Largest Prime Factors …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. $y=\frac{&space;\left&space;(&space;1+\sqrt{2}&space;\right&space;)^{n}-\left&space;(&space;17+12\sqrt{2}&space;\right&space;)&space;\left&space;(&space;1-\sqrt{2}&space;\right&space;)^{n}&space;}{4+3\sqrt{2}}+2$ …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. y = 341872011 … 072849012 (n = 299) 114 269914496787930788048518982299248090651233 {42, P} • 25033433662289323209446285522381470904942857 {44, P} Kyriakos Kefalas 10.4.2019 y = 943894 … 089833 (n = 401) 153 239917951276424996019126892293697275235155019201 {48, P} • 1362206454708311709881839401901924110767184969713808434893286849 {64, P} Kyriakos Kefalas 10.4.2019 …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. $y=\frac{&space;\left&space;(&space;1+\sqrt{2}&space;\right&space;)^{n}-\left&space;(&space;17+12\sqrt{2}&space;\right&space;)&space;\left&space;(&space;1-\sqrt{2}&space;\right&space;)^{n}&space;}{4+3\sqrt{2}}+4$ …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. y = 502935918 … 863265444 (n = 626) 239 3415198373946260820484770872077705324157181395263586097270247531636190\ 68555057642220752753097 {93, P} • 5441129188648159557123888193326449759428192410017861032498419761606401\ 83319419673484290047316659478308640045481 {111, P} Kyriakos Kefalas 15.4.2019 …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. $y&space;=&space;2^{n}+31^{n}$ …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. y = 175955193 … 703002783 (n = 261) 390 1056467878812408725826840484801176448929316908412493221155137773092027\ 25528695973 {81, P} • 88597062218909663983360542280022237915181715973729402116124741597814397\ 7861501952827552396669957874864659903109975159822272004075363973506283\ 9787910221002199344854697409363230221149982051106056360643038509837507\ 7865651 {218, P} Kyriakos Kefalas 1.5.2019 y = 465218884 … 961833025 (n = 270) 403 1840162452318481448279440586720262762082487884501823259127863781 {64, P} • 39346611859851985292525094222317007866711926793330020742265225600268114\ 8118019908847684846676896217743553848047158600949490247475269437679646\ 00159132927198090462431666084448559523584606877126768399671784190321 {209, P} Kyriakos Kefalas 1.5.2019 y = 256158133 … 871221377 (n = 300) 448 22661712878000600659849239636403642442210062322990094011917619424701328\ 07243915567206210741355278955653714925801 {112, P} • 4148308185752761126063784095161976756434408816393239073681318674659682927\ 844520237614073912482883564389381754870479613483372053688670835238406156\ 771160042088562302688886483307649086809005624197854977799552662982851743\ 6639208066846611123201 {239, C} Kyriakos Kefalas 1.5.2019 …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. $y&space;=&space;x^{2}+1$ …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. …………….. y =127735204 … 892714162 (x = 113020000000000000000000\ 0000015822900000000000000\ 0000000000813753000000000\ 0000000000000018535540000\ 00000000000000000001650119 {125, P} 249 21381437576679180509365711102012121564590970961014369608939322473558645220\ 7996914230928968726226514094429836171149 {114, P} • 1277352040000000000000000000017883154600000000000000000000091971381250\ 00000000000000000020949161164000000000000000000018649950097  {129, P} Kyriakos Kefalas 9.5.2019 …………………………………………….. ……………. ……………………………………………………………………………………………………………………………………….. ……………….. ……………..

Created 10.4.2019