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 =
 | 
– 7 x
– 17 = | |
|---|
| 119 | |
|---|
| | + 7 x
+ 17 =
|
– 11 x
– 13 = | |
|---|
| 143 | |
|---|
 | SOPF = 24
This  triplet has been found [ September, 2000 ]
by Patrick De Geest | |
|---|
| LENGTH 6
174191 | |
|---|
| | + 373 x
+ 467 =
|
– 383 x
– 457 = | |
|---|
| 175031 | |
|---|
| | + 383 x
+ 457 =
|
– 397 x
– 443 = | |
|---|
| 175871 | |
|---|
| SOPF = 840
This  triplet has been found [ September, 2000 ]
by Patrick De Geest | |
|---|
| LENGTH 9
298687992 | |
|---|
| | + 2 . 2 . 2 . 3 . 179 . 251 . 277 =
|
– 2 . 2 . 17 . 53 . 179 . 463 = | |
|---|
| 298688708 | |
|---|
| | + 2 . 2 . 17 . 53 . 179 . 463 =
|
– 2 . 2 . 2 . 2 . 11 . 19 . 179 . 499 = | |
|---|
| 298689424 | |
|---|
| SOPF = 716
This  triplet has been found [ September, 2000 ]
by Patrick De Geest | |
|---|
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
| LENGTH 22
2479459309123932072479 | |
|---|
| | + 49785637921 x
+ 49802702399 =
|
– 49785643759 x
– 49802696561 = | |
|---|
| 2479459309223520412799 | |
|---|
| | + 49785643759 x
+ 49802696561 =
|
– 49785649601 x
– 49802690719 = | |
|---|
| 2479459309323108753119 | |
|---|
| SOPF = 99588340320
Triplet found on [ October 13, 2000 ]
by Roberto Botrugno | |
|---|
| [RECORD] LENGTH 102
640000028235486939877676616625399472546557165146838/
039766449055866620240288557497048899726433358934239 | |
|---|
| | +
80000001764717913735538026/
1415804098210232705451361
+
80000001764717914821305088/
7468143253234671888129599
=
|
=
–
80000001764717914821305088/
7468143105873410858968081
–
80000001764717913735538026/
1415804245571493734612879 | |
|---|
| 640000028235486939877676616625399472546557165146839/
639766484350224905808719706380996251171337952515199 | |
|---|
| | +
80000001764717914821305088/
7468143105873410858968081
+
80000001764717913735538026/
1415804245571493734612879
=
|
=
–
80000001764717913735538026/
1415804392932754763774401
–
80000001764717914821305088/
7468142958512149829806559 | |
|---|
| 640000028235486939877676616625399472546557165146841/
239766519644583191377150855264943602616242546096159 | |
|---|
| SOPF = 1600000035294358285568431148883947351444904593580960
Triplet submitted on [ January 13, 2002 ]
by Roberto Botrugno | |
|---|
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. )
|
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.