Problems & Puzzles: Puzzles

Puzzle 918. Primes on the line y=x

Vic Bold sent the following nice puzzle.

Let's  make the coordinates (X, Y) of a point in the Cartesian space to be related with the primes numbers P, this way:

Xi=P(2*i-1) and Yi=P(2*i), for i=>1..... Def. 1

Accordingly with this definition (Def. 1):

 

i Xi Yi
1 2 3
2 5 7
3 11 13
4 17 19
5 23 29
6 31 37
... ... ...

This means that certain class of couples of consecutive primes define a point in the Cartesian space.

Vic Bold looks for points that are on a line straight and parallel to the line y=x, and more specifically asks how many k consecutive points as these could you find.

He provided two examples:

a) k=4. The first four consecutive points as these are:

(97, 101), (103, 107), (109,113,), (127,131)

b) k=5. The first five consecutive points as these are:

(853,857), (859,863), (877,881), (883,887), (907,911)

I (CR) found the first example for k=6. The first six consecutive points as these are:

(3225534199, 3225534203), (3225534223, 3225534227),
(3225534229, 3225534233), (3225534289, 3225534293),
(3225534313, 3225534317), (3225534349, 3225534353)

 

Q1. Please find the first examples, for k>6

Q2. Redo Q1 if you change Def. 1 in order to let out the prime 2, that is to say, to include only the odd primes.

 

 

Introducing note:

I apologize because for some few hours I was asking an improper question, "points on the line y=x". Fortunately Jan van Delden advised to me about this mistake and I switched the question for "points on a parallel line to y=x".

Despite that the two Vic examples and my own search was for points over certain parallel line to y=x, specifically y=x+4, I let the questions Q1 & Q2 of this puzzle open for points over the line parallel to y=x namely y=x+b, without any restriction about the b value.

***

Contributions came from Jan van Delden, Jeff Heleen, Emmanuel Vantieghem and Giovanni Resta.

***

Jan wrote:

Solutions necessarily have the form (p,p+a)(q,q+a) etc.

 

Q1:

 

k= 6

(6311,6317)(6323,6329)(6337,6343)

(6353,6359)(6361,6367)(6373,6379)

 

k=7

(5880031,5880037)(5880041,5880047)(5880053,5880059)(5880067,5880073)

(5880103,5880109)(5880137,5880143)(5880163,5880169)

 

k=8

(17804993,17804999)(17805017,17805023)(17805031,17805037)

(17805083,17805089)(17805091,17805097)(17805113,17805119)

(17805143,17805149)(17805157,17805163)

 

k=9

(368187943,368187949)(368187971,368187977)(368187997,368188003)

(368188021,368188027)(368188033,368188039)(368188043,368188049)

(368188063,368188069)(368188081,368188087)(368188097,368188103)

 

k=11

(243583588177,243583588207)(243583588249,243583588279)(243583588369,243583588399)

(243583588489,243583588519)(243583588663,243583588693)(243583588699,243583588729)

(243583588753,243583588783)(243583588807,243583588837)(243583588933,243583588963)

(243583589017,243583589047)(243583589059,243583589089)

 

Q2:

 

k=3

(7,11)(13,17)(19,23)

 

k=4

(13499,13513)(13523,13537)(13553,13567)(13577,13591)

 

k=5

(106357,106363)(106367,106373)(106391,106397)

(106411,106417)(106427,106433)

 

k=6

(5950837,5950843)(5950871,5950877)(5950891,5950897)

(5950921,5950927)(5950943,5950949)(5950957,5950963)

 

k=8

(43759481,43759487)(43759517,43759523)(43759531,43759537)

(43759543,43759549)(43759553,43759559)(43759607,43759613)

(43759621,43759627)(43759657,43759663)

 

k=9

(77193802909,77193802939)(77193802951,77193802981)(77193803023,77193803053)

(77193803059,77193803089)(77193803299,77193803329)(77193803353,77193803383)

(77193803401,77193803431)(77193803773,77193803803)(77193804223,77193804253)

 

k=11

(2047469852497,2047469852527)(2047469852569,2047469852599)(2047469852689,2047469852719)

(2047469852809,2047469852839)(2047469852983,2047469853013)(2047469853019,2047469853049)

(2047469853073,2047469853103)(2047469853127,2047469853157)(2047469853253,2047469853283)

(2047469853337,2047469853367)(2047469853379,2047469853409)

 

Values for k that are missing can be given, but this requires a little tweaking.

 

Note I also tried to find solutions of the form (p,p+a,p+a+b)(q,q+a,q+a+b), itís 3-dimensional equivalent.

As can be expected (larger) solutions are harder to find.

 

Starting at (2,3,5):

 

k=2

(1949,1973,1979)(2039,2063,2069)

 

k=3

(2665031,2665043,2665109)(2665121,2665133,2665199)(2665241,2665253,2665319)

 

k=4

(2009844701,2009844713,2009844743)(2009844779,2009844791,2009844821)

(2009844911,2009844923,2009844953)(2009844989,2009845001,2009845031)

 

k=5

(188225534519,188225534561,188225534579)(188225534639,188225534681,188225534699)

(188225534747,188225534789,188225534807)(188225534849,188225534891,188225534909)

(188225535077,188225535119,188225535137)

 

Starting at (3,5,7):

 

k=2

(59,61,67)(71,73,79)

 

k=3

(139291,139297,139301)(139303,139309,139313)(139333,139339,139343)

 

k=4

(1618801753,1618801759,1618801771)(1618801781,1618801787,1618801799)

(1618801813,1618801819,1618801831)(1618801861,1618801867,1618801879)

 

k=5

(288501238709,288501238763,288501238793)(288501238829,288501238883,288501238913)

(288501238919,288501238973,288501239003)(288501239099,288501239153,288501239183)

(288501239249,288501239303,288501239333)

 

Starting at (5,7,11):

 

k=2

(61,67,71)(73,79,83)

 

k=3

(1268747,1268753,1268759)(1268777,1268783,1268789)(1268791,1268797,1268803)

 

k=4

(148530307,148530311,148530329)(148530337,148530341,148530359)

(148530397,148530401,148530419)(148530439,148530443,148530461)

 

k=5

(18556024163,18556024187,18556024247)(18556024313,18556024337,18556024397)

(18556024427,18556024451,18556024511)(18556024637,18556024661,18556024721)

(18556024817,18556024841,18556024901)

***

Jeff wrote, on March 28, 2018:

I found an earlier solution for k=6.

Plus k=7 , k=8 and k=9.

 

k=6

(6311, 6317)

(6323, 6329)

(6337, 6343)

(6353, 6359)

(6361, 6367)

(6373, 6379)

 

k=7

(5880031, 5880037)

(5880041, 5880047)

(5880053, 5880059)

(5880067, 5880073)

(5880103, 5880109)

(5880137, 5880143)

(5880163, 5880169)

 

k=8

(17804993, 17804999)

(17805017, 17805023)

(17805031, 17805037)

(17805083, 17805089)

(17805091, 17805097)

(17805113, 17805119)

(17805143, 17805149)

(17805157, 17805163)

 

k=9

(368187943, 368187949)

(368187971, 368187977)

(368187997, 368188003)

(368188021, 368188027)

(368188033, 368188039)

(368188043, 368188049)

(368188063, 368188069)

(368188081, 368188087)

(368188097, 368188103)

 

***

Emmanuel wrote on March 29, 2018

I searched  k  consecutive points that are on some line of the form y = x + b, where  b may depend on the value of  k.
 
Q1. (start point  (2, 3).
k = 3 :
 (5, 7), (11, 13), (17, 19)  all on the line  y = x + 2
k = 4 :
 (97, 101), (103, 107), (109, 113), (127, 131)  all on the line  y = x + 4
k = 5 :
 (853, 857), (859, 863), (877, 881), (883, 887), (907, 911)  all on the line  y = x + 4
k = 6 :
(6311, 6317), (6323, 6329), (6337, 6343), (6353, 6359), (6361, 6367), (6373, 6379)
  all on the line  y = x + 6, will be the case in the next three examples :
k = 7 : 
(5880031, 5880037), (5880041, 5880047), (5880053, 5880059), (5880067, 5880073), (5880103, 5880109), (5880137, 5880143), (5880163, 5880169)
k = 8 :
(17804993, 17804999), (17805017, 17805023), (17805031, 17805037), (17805083, 17805089), (17805091, 17805097), (17805113, 17805119), (17805143, 17805149), (17805157, 17805163)
k = 9 :
(368187943, 368187949), (368187971, 368187977), (368187997, 368188003), (368188021, 368188027), (368188033, 368188039), (368188043, 368188049), (368188063, 368188069), (368188081, 368188087), (368188097, 368188103)
 
Q2. (start point  (3, 5)
k = 3 :
(7, 11), (13, 17), (19, 23)  all on the line  y = x + 4
k = 4 :
(13499, 13513), (13523, 13537), (13553, 13567), (13577, 13591))  all on the line  y = x + 14
k = 5 :
(106357, 106363), (106367, 106373), (106391, 106397), (106411, 106417), (106427, 106433)  all on the line  y = x + 6
k = 6 :
(5950837, 5950843), (5950871, 5950877), (5950891, 5950897), (5950921, 5950927), (5950943, 5950949), (5950957, 5950963)  all on the line  y = x + 6  which is the same line for the next  k  points
k = 7 :
(43759481, 43759487), (43759517, 43759523), (43759531, 43759537), (43759543, 43759549), (43759553, 43759559), (43759607, 43759613), (43759621, 43759627)
k = 8 :
(43759481, 43759487), (43759517, 43759523), (43759531, 43759537), (43759543, 43759549), (43759553, 43759559), (43759607, 43759613), (43759621, 43759627), (43759657, 43759663)
k = 9 :
(6519095747, 6519095753), (6519095767, 6519095773), (6519095777, 6519095783), (6519095813, 6519095819), (6519095831, 6519095837), (6519095851, 6519095857), (6519095881, 6519095887), (6519095911, 6519095917), (6519095923, 6519095929)

***

Giovanni wrote on March 30, 2018

For the first part I got these starting primes for record runs of collinear points


6: 6311
7: 5880031
8: 17804993
9: 368187943
10: 324838025843
11: 16475123840867

for the second part:

4: 13499
5: 106357
6: 5950837
8: 43759481
9: 43597633337
10: 3367430253977
11: 109953426948341

***

 

 


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