WON TOPIC 222

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[ October 9, 2022 ] [ Last update September 18, 2023 ]
Highlighting semiprime palindromes with record merit scores.

Patrick De Geest Merit Score 0,987654...

Let us define this merit score (approach by courtesy of Alexandru Petrescu).
Let P = p₁ * p₂ P = palindrome ; p₁ & p₂ prime factors ; p₁ < p₂

MS = \( {{2\ *\ p_1} \over {p_1\ +\ p_2}}\)

As I am considering only sporadic semiprimes I will leave out the
palindromic squares, or the cases whereby p₁ = p₂, which always
yield the top merit score of 1. So the upper bound becomes 0,999999...

To avoid the trivial cases I will consider only palindromes with lengths ⩾ 5.

Some large semiprime palindromes will never reach the top of the list with best
merit scores. Yet due to their sizes they have an impressive appearance.
In annex to the list I will show some examples without bothering with merit
and ordered by the size of the smallest primefactor p₁.

Let me show you for each palindromic numberformat (SUPP, PWP & PDP)
a selection whereby the length of p₁ is well over half the length of p₂.
When both p₁ and p₂ have lengths ⩾ 100 the highlights are in yellow .

The finest semiprime SUPP's

(38*10^5-83)/99 { p3 * p3 } 131 * 293  MS = 0,617925...

(38*10^29-83)/99 { p15 * p15 } 127342353197113 * 301422405194791  MS = 0,593996...

(74*10^35-47)/99 { p17 * p18 } 80905830103195001 * 923882427905809747  MS = 0,161040...

(79*10^9-97)/99 { p4 * p6 } 7187 * 111031  MS = 0,121589...

(98*10^15-89)/99 { p7 * p9 } 5923097 * 167125237  MS = 0,068456...

(98*10^11-89)/99 { p5 * p7 } 48821 * 2027609  MS = 0,047024...

(19*10^25-91)/99 { p12 * p14 } 174551368897 * 10994997812503  MS = 0,031254...

•••

\( {{(98*10^{99}-89)} \over 99} \) {p39} 260156792519156427914594659296032226221
{p61} 3805009203540589909709562711595275004238564354139790637203009

\( {{(98*10^{89}-89)} \over 99} \) {p42} 559734928538684225342816557954109919167929
{p48} 176851387938813668219697246806390393621820349141

\( {{(78*10^{107}-87)} \over 99} \) {p44} 51618316189837999713481301122463343843497743
{p64} 1526355073228630587726658399348398542297837925279399823133253709

The finest semiprime PWP's

(10^7+36*10^3-1)/9 { p4 * p4 } 1051 * 1061  MS = 0,995265...

10^9-5*10^4-1 { p5 * p5 } 30109 * 33211  MS = 0,951011...

10^5-*10^2-1 { p3 * p3 } 283 * 353  MS = 0,889937...

10^43-7*10^21-1 { p22 * p22 } 2215006022870241309047 * 4514660410287207794617  MS = 0,658281...

10^11-7*10^5-1 { p6 * p6 } 205537 * 486527  MS = 0,593983...

(10^11+27*10^5-1)/9 { p5 * p6 } 59957 * 185323  MS = 0,488886...

10^5-4*10^2-1 { p3 * p3 } 137 * 727  MS = 0,317130...

10^5-7*10^2-1 { p3 * p3 } 109 * 911  MS = 0,213725...

(7*10^5-54*10^2-7)/9 { p2 * p4 } 71 * 1087  MS = 0,122625...

•••

\( {(10^{57}-5*10^{28}-1)} \)

{p28} 3404730839554091347889762281
{p30} 293708973520787144658793555879

Source

\( {(10^{75}-8*10^{37}-1)} \)

{p32} 70490999036439981131047105940143
{p44} 14186208362333676601114110453273238264066993

Source

\( {{(7*10^{103}+36*10^{51}-1)} \over 9} \)

{p37} 8963142674648807697594387240247111949
{p66} 123964456602231479489609870355391974237440495595831652047932059939

Source

\( {{(10^{93}+45*10^{46}-1)} \over 9} \)

{p39} 363639540323592843973557326487709158203
{p54} 305552886279188406342444476482262011754545879096939237

Source

\( {{(10^{107}+54*10^{53}-1)} \over 9} \)

{p45} 667270284341444404211801866163691905364260573
{p62} 16651589875723461599281172074878426240204490561530052480382707

Source

\( {(10^{105}-2*10^{52}-1)} \)

{p46} 1863079667124649535843265309290442882620142287
{p60} 536745699953524798805570660299784549337566683355200355789777

Source

\( {{(7*10^{109}-54*10^{54}-7)} \over 9} \)

{p52} 1199703120496327971166912328502612111741137333287561
{p58} 6483085394126541203968119534230642078504648703049343858857

Source

\( {{(7*10^{115}-18*10^{57}-7)} \over 9} \)

{p52} 4908080039210391222640407293380788169196549342459237
{p64} 1584688455697854041471015089623685838814364850834967536965947421

Source

\( {{(10^{139}+63*10^{69}-1)} \over 9} \)

{p57} 467652694300191693736261084612152646516775813624373744143
{p82} 2375932234868887528151906009973851128781214167960123415164812510915805241727384777

Source

\( {{(7*10^{131}-27*10^{65}-7)} \over 9} \)

{p64} 2329957521561874313076509630588577468145156433739855730454265377
{p68} 33381629088945736991421982152696814983525014160319979864132043781201

Source

\( {{(10^{159}+63*10^{79}-1)} \over 9} \)

{p67} 4962465444058938655391243644518305506591809851513115319410780313547
{p92} 22390304247686657883143862633271938626677676998370398726721461608016405878157493281781905013

Source

\( {10^{169}-10^{84}-1)} \)

{p69} 100036320130834943529138780664396063768786416272340778924697781016049
{p101} 99963693055894657701001596134073004248220745363699422403371443828602899053089704053788701792852493551

Source

\( {{(7*10^{187}-18*10^{93}-7)} \over 9} \)

{p70} 1293522756995720816564602491978997805189830830264184295379328736337533
{p118} 6012865050663741766935035576949502728047778898867929764946741465928976131662651136973092902784709755212981766002136069

Source

\( {(10^{187}-8*10^{93}-1)} \)

{p80} 50019775114615199495075917230691916547419255646795574679062925400535109003749027
{p108} 199920930813583680285643755589862706279286424969580548098469179485993277480268624034902644118801184584418037

Source

\( {{(7*10^{193}-54*10^{96}-7)} \over 9} \)

{p82} 7561360046173779778582655360183561408364806872456775270433495743212570624432911847
{p112} 1028621535052217265514280258883802609929861144866809713484077027862306680278819515852994142689642991126705545191

Source

\( {{(10^{175}+15*10^{87}-1)} \over 3} \)

{p83} 18631732445417539147677250699850654482568802964384935829186788219738329283320204819
{p93} 178906247344334541902715933521072667057137724490133579473554509745153470543910096245616188407

Source

\( {{(10^{217}+6*10^{108}-1)} \over 3} \)

{p97} 1477905086797375456700160945763956990650026329957508704705714027637105295419187796395742826423139
{p121} 2255444793519640821534197987432315516647266790013197787910910258778536727335201523399201377417287175529222940314399862247

Source

\( {(10^{253}-7*10^{126}-1)} \)

{p117} 211857628677636264166676902114066256198845941934180788645905936104472665184004014226679863567582409810768749673482061
{p137} 47201510100993602326827859852217555163129145401663287219390072454758015933440888931227171966024510895342063068118093723212737563878094459

Source

\( {{(10^{241}+63*10^{120}-1)} \over 9} \)

{p119} 12530152577707140852678607904315228208483149896115932457293358540343382941957056668184690807815356762378031023874074769
{p122} 88674986535114512159176731810001756112004440722794152102533707683005633552791743573754986450421838115667305782337688674519

Source

The finest semiprime PDP's

(4*10^8-7)/3 { p5 * p5 } 11287 * 11813  MS = 0,977229...

8*10^10-3 { p6 * p6 } 164113 * 487469  MS = 0,503737...

2*10^16-9 { p8 * p9 } 73281367 * 272920673  MS = 0,423345...

(89*10^26+1)/9 { p14 * p14 } 15090452766973 * 65530763334893  MS = 0,374354...

(4*10^56-7)/3 { p28 * p29 } 8992981486986397211268073 * 1482637693920285654052934049744347  MS = 0,104557...

•••

\( {{(34*10^{56}-43)} \over 9} \)

{p27} 360923792711553185317556497
{p31} 1046696796959833941452450026909

Source

\( {{(71*10^{88}-17)} \over 9} \)

{p38} 13462413365458307618251578178697597023
{p52} 5859936606262664353997676487761580940071396500304169

Source

\( {(8*10^{90}-3)} \)

{p42} 410080876317578982031342902685566416497267
{p50} 19508346918875970808229263506124178004728128266191

Source

\( {{(16*10^{84}-61)} \over 9} \)

{p42} 756600723901820421121949766505399991678987
{p43} 2349690823198933964135445765680242530617633

Source

MS = 0,487140...

\( {{(82*10^{104}+71)} \over 9} \)

{p46} 2877334510774662743911313907879097392207174547
{p60} 316651090688031718750771928740237068788953977912411481493077

Source

\( {(2*10^{104}-9)} \)

{p47} 58246367832510471330922533301678866841362759833
{p58} 3433690501957258584231539706004296854340296493847755296527

Source

\( {(2*10^{116}-9)} \)

{p58} 1300077873236814617625875427828886902290588172083356016807
{p60} 153836938630497839571709555448490510158389234978218219454513

Source

\( {{(65*10^{152}+43)} \over 9} \)

{p67} 6550213397685364775986550762989701971314721157128662740183066777671
{p87} 110259342463168051908664963775205452537386095869746869361395515038990560570276567780437

Source

\( {{(34*10^{150}-43)} \over 9} \)

{p73} 2980537215871294317759980094532234409121960036227028511662295660263806351
{p79} 1267482169879038992405294059376824542448418727805469492688123456470418239686723

Source

\( {{(5*10^{160}-17)} \over 3} \)

{p73} 7882418625046493137079200109466142233457102918797363849268660188394441163
{p88} 2114410241256167881019929854171578797976297286873491215535819386209922986237183824564047

Source

\( {{(32*10^{108}-23)} \over 9} \)

{p77} 22505468127454808181216964200487999120781593821049054585465704599969960629843
{p111} 157986296282283153038019270956995707149770776451983434848501382962868901158389576269567185495417338509918446971

Source

\( {{(23*10^{174}+1)} \over 3} \)

{p82} 1160607580637197436909970824725037202415122085752327758687078618700131481703406693
{p94} 6605735473877836429266158816256245412965621347106431956774582202198506986337160850347488301519

Source

\( {{(5*10^{198}-17)} \over 3} \)

{p90} 119210534379624869888380873328590164795408717116498507216021846495878475022485775217297333
{p110} 13980867339787117486389823283545379889344751436236630206822306072817458470959671454043114394014893486370164017

Source

“ALLSORTS” semiprime palindromes

This last section is for the motivated reader.
If you constructed a semiprime palindrome that is not of a known
format i.e. 'sporadic' like the above ones then it will find a
place in this table. The longer the semiprimes and the closer
their lengths are to each other the better the chance it will end
up high in the ranking. Tell me how you searched for it as well.

  {sp21} 180163391219193361081 {p11} 13422495703
{p11} 13422495727   Note: two consecutive primes
  MS = 0,9999999991059...

  {PG7 index nr 76} {p25} 1159363692769172939242993
{p25} 2898409231922932348107481   A palindromic heptagonal with two equal length primefactors
  MS = 0.571428...

  {#1669} 10001001001010010010001 {p11} 24847181327
{p12} 402500423263   Basenumber of Palindromic Square (BL=23)
  MS = 0,116285...

•••

Sources on the Internet around Semiprimes

Wolfram MathWorld : Semiprime
Wikipedia : Semiprime
PrimePages : Glossary - semiprime
A001358 - semiprimes: product of two primes
A078972 - Brilliant numbers: semiprimes whose prime factors have the same number of decimal digits
Brilliant numbers by Dario Alpern
sidef-scripts/Math/brilliant_numbers_count.sf