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Sets of three composites in bi-directional
'Sum Of Prime Factors' pro-/ retrogression
rood comments



Introduction

This page is the continuation of the topic started in WONplate 78.
Remarkable progress was made by Roberto Botrugno from Italy.
Please read on for a detailed explanation of his method
and enjoy the larger solutions he discovered using his algorithm.


The Tables


LENGTH 2/3
95
 + 5 x
+ 19 =
down
up
7 x
17 =
 
119
 + 7 x
+ 17 =
down
up
11 x
13 =
 
143
!SOPF = 24
This triplet has been found [ September, 2000 ]
by Patrick De Geest
!

brown line

LENGTH 6
174191
 + 373 x
+ 467 =
down
up
383 x
457 =
 
175031
 + 383 x
+ 457 =
down
up
397 x
443 =
 
175871
!SOPF = 840
This triplet has been found [ September, 2000 ]
by Patrick De Geest
!

brown line

LENGTH 9
298687992
 + 2 . 2 . 2 . 3 . 179 . 251 . 277 =
down
up
2 . 2 . 17 . 53 . 179 . 463 =
 
298688708
 + 2 . 2 . 17 . 53 . 179 . 463 =
down
up
2 . 2 . 2 . 2 . 11 . 19 . 179 . 499 =
 
298689424
!SOPF = 716
This triplet has been found [ September, 2000 ]
by Patrick De Geest
!

brown line

Despite the extraordinary efforts of Roberto Botrugno
( please take a look at his remarkable solutions of up to length 62 !! )
the FOURTH triplet still hasn't been determined.

The following triplet is the smallest one whereby the terms have exactly two prime factors
immediately followed by his record triplet with terms of length 102 !
In the words of Roberto Botrugno :

" I have verified every possible combination between sopf_840 and sopf_99588340320
(my smallest solution) and there aren't any other P_sopf solutions.
I don't know if there is a F_sopf."

Definition of P-sopf and F_sopf.
F_sopf is a number that creates more than 2 factors numbers in every term (sopf_716).
P_sopf is a number that creates only 2 prime factors for every term (sopf_24, sopf_840, ...).
At the moment to find P_sopf is easier for me than to find F_sopf
so I'll work on the F_sopf ehehe :-)


Read on for a more detailed explanation of his P_sopf algorithm

brown line

LENGTH 22
2479459309123932072479
 + 49785637921 x
+ 49802702399 =
down
up
49785643759 x
49802696561 =
 
2479459309223520412799
 + 49785643759 x
+ 49802696561 =
down
up
49785649601 x
49802690719 =
 
2479459309323108753119
!SOPF = 99588340320
Triplet found on [ October 13, 2000 ]
by Roberto Botrugno
!

[RECORD] LENGTH 102
640000028235486939877676616625399472546557165146838/
039766449055866620240288557497048899726433358934239 
 +
80000001764717913735538026/
1415804098210232705451361
+
80000001764717914821305088/
7468143253234671888129599
=
down
up
=

80000001764717914821305088/
7468143105873410858968081

80000001764717913735538026/
1415804245571493734612879
 
640000028235486939877676616625399472546557165146839/
639766484350224905808719706380996251171337952515199 
 +
80000001764717914821305088/
7468143105873410858968081
+
80000001764717913735538026/
1415804245571493734612879
=
down
up
=

80000001764717913735538026/
1415804392932754763774401

80000001764717914821305088/
7468142958512149829806559
 
640000028235486939877676616625399472546557165146841/
239766519644583191377150855264943602616242546096159 
!SOPF = 1600000035294358285568431148883947351444904593580960
Triplet submitted on [ January 13, 2002 ]
by Roberto Botrugno
!

brown line

Roberto Botrugno made a connection with Puzzle 97 of PP&P
[ January 13, 2002 ]

Note that

298687992/716 = 417162
298688708/716 = 417163
298689424/716 = 417164

if

1) sopf(a) = sopf(a+1) = sopf(a+2) ;
2) the equation sopf(a)+b+c = b*c has integer solutions
3) sopf(a*b*c) = sopf(a*b*c+1)
then
a*b*c is a F_sopf.

Example

533+(535+3) = 535*2 but sopf(417162*535*2) = 647
533+(268+3) = 268*3 but sopf(417162*268*3) = 607
533+(90+7) = 90*7 but sopf(417162*90*7) = 553
533+(179+4) = 179*4 and sopf(417162*179*4) = sopf(417163*179*4)

Maybe four consecutive numbers with the same sopf create a quadrisopf
of the form F_sopf.

(ps. quadrisopf of the form P_sopf is impossible because there doesn't
exist 4 squares in arithmetic progression. )

brown line

Here is Roberto Botrugno's method for discovering these huge P_sopf solutions


First example consider sopf_24 (95, 119, 143). 

Note that :
    95 + 24 + 24 = 143
    95 + 24 + 24 + 1 = 144
    95 + 49 = 12^2
    95 + 7^2 = 12^2

    119 + 24 = 143
    119 + 24 + 1 = 144
    119 + 25 = 12^2
    119 + 5^2 = 12^2

    143 + 1 = 144
    143 + 1 = 12^2
    143 + 1^2 = 12^2

Rewrite (95, 119, 143) as follows
    (12^2-7^2, 12^2-5^2, 12^2-1^2)
Note that 1^2, 5^2 and 7^2 are squares in arithmetic progression (24).

    a^2 - b^2 = (a+b)(a-b)
    (12+7)(12-7) = 95,
    (12+5)(12-5) = 119,
    (12+1)(12-1) = 143.

sopf_((12+7)+(12-7)) = sopf_((12+5)+(12-5)) = sopf_((12+1)+(12-1)) = 24
if and only if
12+-7, 12+-5 and 12+-1 are all primes.

Counterexample consider 60^2-17^2, 60^2-13^2 and 60^2-7^2 =>
    77 * 43,  47 * 73 , 67 * 53.
giving
    77 * 43 = 3311
    47 * 73 = 3431
    67 * 53 = 3551
and
    3551 - 3431 = 120
    3431 - 3311 = 120
but, sopf_(77 * 43) = 61 because 77 is composite.
sopf_(47 * 73) = 120 and sopf_(67 * 53) = 120 because 47 and 67
are all prime.
17^2, 13^2 and 7^2 are square numbers in arithmetic progression (120).

Second example consider sopf_840 (174191, 175031, 175871)
These triplet terms can be expressed as a difference of 2 squares :
    174191 = 420^2 - 47^2
    175031 = 420^2 - 37^2
    175871 = 420^2 - 23^2

420+-47=373,467
420+-37=383,457
420+-23=397,443

sopf_(467 * 373) = sopf_(457 * 383) = sopf_(443 * 397) = 840
because all six resulting numbers are prime.
47^2, 37^2 and 23^2 are in arithmetic progression (840).

In general the problem is to find 3 square numbers in
arithmetic progression so that
a^2-b^2=k , b^2-c^2=k , (e.g. 47^2-37^2=840, 37^2-23^2=840)
and (k/2)+-a, (k/2)+-b, (k/2)+-c are prime.

Consider 3 integers a,b,c.
If a^2-b^2=k and b^2-c^2=k
    then a^2, b^2 and c^2 are 3 square numbers in arithmetic
    progression with difference k.
If k/2+-a, k/2+-b and k/2+-c are all primes
    then (k/2)^2-a^2,(k/2)^2-b^2 and(k/2)^2-c^2 are 3 numbers
    in arithmetic progression
    with difference k having the same sopf = k.






Enjoy the following P_sopf solutions kindly sent to me by Roberto !
It is not an exhaustive list. Roberto has in the mean time found many more triplets.
If you want an up_to_date list please contact Roberto himself.

brown line

LENGTH 37
4478902218446180230475773337554877471
 + 2116340668031034373 x
+ 2116342744863087827 =
down
up
2116340668033072423 x
2116342744861049777 =
 
4478902218446180234708456750448999671
 + 2116340668033072423 x
+ 2116342744861049777 =
down
up
2116340668035110477 x
2116342744859011723 =
 
4478902218446180238941140163343121871
!SOPF = 4232683412894122200
Triplet found on [ October 14, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 41
59336364903335477410849865182584799047599
 + 243590543603140562401 x
+ 243590592744875609999 =
down
up
243590543603150476199 x
243590592744865696201 =
 
59336364903335477411337046318932815219999
 + 243590543603150476199 x
+ 243590592744865696201 =
down
up
243590543603160390001 x
243590592744855782399 =
 
59336364903335477411824227455280831392399
!SOPF = 487181136348016172400
Triplet found on [ October 12, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 42
119883596739565730384906393185208734423871
 + 346242138177475941043 x
+ 346242076052904028997 =
down
up
346242076052915175703 x
346242138177464794337 =
 
119883596739565730385598877399439114393911
 + 346242076052915175703 x
+ 346242138177464794337 =
down
up
346242076052926322413 x
346242138177453647627 =
 
119883596739565730386291361613669494363951
!SOPF = 692484214230379970040
Triplet found on [ October 17, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 44
99861850740473507956819908315455767890016079
 + 9993090441989232208399 x
+ 9993089857455040880321 =
down
up
9993089857455075071959 x
9993090441989198016761 =
 
99861850740473507956839894495755212163104799
 + 9993089857455075071959 x
+ 9993090441989198016761 =
down
up
9993089857455109263601 x
9993090441989163825119 =
 
99861850740473507956859880676054656436193519
!SOPF = 19986180299444273088720
Triplet found on [ October 17, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 45
383709604779964076470710078188419776920148431
 + 19588507414145606823347 x
+ 19588506498605032320373 =
down
up
19588506498605117902637 x
19588507414145521241083 =
 
383709604779964076470749255202332527559292151
 + 19588506498605117902637 x
+ 19588507414145521241083 =
down
up
19588506498605075111503 x
19588507414145564032217 =
 
383709604779964076470788432216245278198435871
!SOPF = 39177013912750639143720
Triplet found on [ October 17, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 45 bis
823681257750517489279343257354195600449721871
 + 28699848286115875767923 x
+ 28699847105079985106677 =
down
up
28699847105080033707823 x
28699848286115827166777 =
 
823681257750517489279400657049586796310596471
 + 28699847105080033707823 x
+ 28699848286115827166777 =
down
up
28699847105080082308973 x
28699848286115778565627 =
 
823681257750517489279458056744977992171471071
!SOPF = 57399695391195860874600
Triplet found on [ October 17, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 47
13553461936419768731101648367275660004006623871
 + 116419338110720381869043 x
+ 116419335106765299516997 =
down
up
116419335106765377027703 x
116419338110720304358337 =
 
13553461936419768731101881205948877489688009911
 + 116419335106765377027703 x
+ 116419338110720304358337 =
down
up
116419335106765454538413 x
116419338110720226847627 =
 
13553461936419768731102114044622094975369395951
!SOPF = 232838673217485681386040
Triplet found on [ October 19, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 47 bis
22315993469127892289058153741682936533594082991
 + 149385387501667542023567 x
+ 149385383954496793627873 =
down
up
149385383954496877855783 x
149385387501667457795657 =
 
22315993469127892289058452512454392697929734431
 + 149385383954496877855783 x
+ 149385387501667457795657 =
down
up
149385387501667373567743 x
149385383954496962083697 =
 
22315993469127892289058751283225848862265385871
!SOPF = 298770771456164335651440
Triplet found on [ October 19, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 47 ter
49134295914902317212587830132553245219420795871
 + 221662574508635370458003 x
+ 221662569893990736829957 =
down
up
221662569893990832899143 x
221662574508635274388817 =
 
49134295914902317212588273457697647845528083831
 + 221662569893990832899143 x
+ 221662574508635274388817 =
down
up
221662574508635178319627 x
221662569893990928968333 =
 
49134295914902317212588716782842050471635371791
!SOPF = 443325144402626107287960
Triplet found on [ October 19, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 47 quater
98562853422203619186223138484174649117621985871
 + 313947217298291184667663 x
+ 313947211478405726881217 =
down
up
313947211478405834768983 x
313947217298291076779897 =
 
98562853422203619186223766378603425814533534751
 + 313947211478405834768983 x
+ 313947217298291076779897 =
down
up
313947211478405942656753 x
313947217298290968892127 =
 
98562853422203619186224394273032202511445083631
!SOPF = 627894428776696911548880
Triplet found on [ October 22, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 48
187135634625938245066744006675771205620668235871
 + 432591768002400607001663 x
+ 432591760795826707125217 =
down
up
432591760795826827179983 x
432591768002400486946897 =
 
187135634625938245066744871859300003847982362751
 + 432591760795826827179983 x
+ 432591768002400486946897 =
down
up
432591760795826947234753 x
432591768002400366892127 =
 
187135634625938245066745737042828802075296489631
!SOPF = 865183528798227314126880
Triplet found on [ October 22, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 48 bis
351659918093807217712934197036534564651474632911
 + 593009210187726022205107 x
+ 593009201294671223104373 =
down
up
593009210187725888840537 x
593009201294671356468943 =
 
351659918093807217712935383054946047048719942391
 + 593009210187725888840537 x
+ 593009201294671356468943 =
down
up
593009201294671489833517 x
593009210187725755475963 =
 
351659918093807217712936569073357529445965251871
!SOPF = 1186018411482397245309480
Triplet found on [ October 24, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 48 ter
941182701119008654205114994468293415076455901871
 + 970145717455670788334483 x
+ 970145705108484848322037 =
down
up
970145705108485005466463 x
970145717455670631190057 =
 
941182701119008654205116934759715979232092558391
 + 970145705108485005466463 x
+ 970145717455670631190057 =
down
up
970145705108485162610893 x
970145717455670474045627 =
 
941182701119008654205118875051138543387729214911
!SOPF = 1940291422564155636656520
Triplet found on [ October 24, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 49
7380882364209115513494986841272831417254107717599
 + 2716777950229649168359999 x
+ 2716777925698789524642401 =
down
up
2716777950229648946861201 x
2716777925698789746141199 =
 
7380882364209115513494992274828707345692800719999
 + 2716777950229648946861201 x
+ 2716777925698789746141199 =
down
up
2716777925698789967640001 x
2716777950229648725362399 =
 
7380882364209115513494997708384583274131493722399
!SOPF = 5433555875928438693002400
Triplet found on [ October 24, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 50
20601686763454138026062787699129695005616575151951
 + 4538908119826166823233587 x
+ 4538908085287073587552373 =
down
up
4538908085287073850379663 x
4538908119826166560406297 =
 
20601686763454138026062796776945900118856985937911
 + 4538908085287073850379663 x
+ 4538908119826166560406297 =
down
up
4538908085287074113206957 x
4538908119826166297579003 =
 
20601686763454138026062805854762105232097396723871
!SOPF = 9077816205113240410785960
Triplet found on [ October 24, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 50 bis
23265215523925629946132294310500868125028601315119
 + 4823402916760346524442899 x
+ 4823402880792671546689781 =
down
up
4823402880792671814897439 x
4823402916760346256235241 =
 
23265215523925629946132303957306665678046672447799
 + 4823402880792671814897439 x
+ 4823402916760346256235241 =
down
up
4823402880792672083105101 x
4823402916760345988027579 =
 
23265215523925629946132313604112463231064743580479
!SOPF = 9646805797553018071132680
Triplet found on [ October 24, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 53
14935926281825707858033173425141254860809482730279471
 + 122212627497103856553949327 x
+ 122212627186823662368971873 =
down
up
122212627186823663156728423 x
122212627497103855766192777 =
 
14935926281825707858033173669566509544737001653200671
 + 122212627186823663156728423 x
+ 122212627497103855766192777 =
down
up
122212627186823663944484977 x
122212627497103854978436223 =
 
14935926281825707858033173913991764228664520576121871
!SOPF = 244425254683927518922921200
Triplet found on [ October 19, 2000 ]
by Roberto Botrugno
!

brown line

LENGTH 54
197738286030718005636905634464690868982085150678971871
 + 444677733147410502428879423 x
+ 444677732413392144559918177 =
down
up
444677732413392145771543823 x
444677733147410501217253777 =
 
197738286030718005636905635354046334542887797667769471
 + 444677732413392145771543823 x
+ 444677733147410501217253777 =
down
up
444677732413392146983169473 x
444677733147410500005628127 =
 
197738286030718005636905636243401800103690444656567071
!SOPF = 889355465560802646988797600
Triplet found on [ November 1, 2000 ]
by Roberto Botrugno
!

brown line

[RECORD] LENGTH 62
64485918110472470752875104339248862702744083677393375856611071
 + 8030312453847311561830726434373 x
+ 8030312454352552222564048871027 =
down
up
8030312454352552222532260802177 x
8030312453847311561862514503223 =
 
64485918110472470752875104339264923327652283541177770631916471
 + 8030312454352552222532260802177 x
+ 8030312453847311561862514503223 =
down
up
8030312453847311561894302572077 x
8030312454352552222500472733323 =
 
64485918110472470752875104339280983952560483404962165407221871
!SOPF = 16060624908199863784394775305400
Triplet found on [ June 10, 2001 ]
by Roberto Botrugno
!






Contributions

Roberto Botrugno (email) from Italy.






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