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Numbers		 HOME plate
WON |

EMBED
Sets of three composites in bi-directional
'Sum Of Prime Factors' pro-/ retrogression
rood WONplate 78


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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