Firstly parse it prime (difference triangle).
It's a name I made up while writing this and I think it's a lousy name. If you can
suggest a better name, please tell me!
In fact - click on
"Prime Reduction Puzzle" or
"Prime Difference Triangle"
to stick a 404 in my web error log and let me know you think it's a better name.
(and if you are a robot, please pretend to click both :-) )
I suggested it as a puzzle on Professor Caldwell's Prime Numbers Mailing List as a silly puzzle. The message can be found at http://groups.yahoo.com/group/primenumbers/message/9046. Here's the important part:
I'm sure everyone's aware of the concept of iteratively taking differences between consecutive terms in sequences of integers. (Ouch, I think I've just made it sound more complicated than it is!) For a finite seqence, each line of differences is one shorter than the one it's below, so that a triangle is formed.
e.g.
1 4 9 16 3 5 7 2 2 0Well, as this is the Primes List, let's throw primes into the mix - I only want to see primes in the triangle. Of course, differences between odd primes will be even, so I'm going to throw a factor of two into the mix too, divide the differences between the above terms by two to form subsequent rows. Note, explicitly, zero and one are _not_ primes. I'm also only interested in positive terms, which means that all of the rows must be strictly increasing.
3 7 2and
5 19 41 7 11 2
I added a variation:
a) What happens if I forbid any prime from being repeated in the triangle?
And if you want to know what specifically we're looking for:
For each triangle size, what's the smallest total sum of all terms?
Sum : 2
2Discoverer : My sentient cactus, Ptolemy, and Lao Tsu.
Sum : 12
3 7 2Discoverer : Phil Carmody, 2002/10/04.
Sum : 71
11 17 31
3 7
2
Discoverer : Phil Carmody, 2002/10/04.
Sum : 294
3 17 43 137
7 13 47
3 17
7
Discoverer : Jack Brennen, 2002/10/04
Sum : 1021
31 37 71 157 431
3 17 43 137
7 13 47
3 17
7
Discoverer : Jack Brennen, 2002/10/04 (verified Minimal J.K. Andersen)
Sum : 6213
59 73 179 433 971 2617
7 53 127 269 823
23 37 71 277
7 17 103
5 43
19
Discoverer : Jack Brennen, 2002/10/04 (verified Minimal J.K. Andersen)
Sum : 30076
919 1013 1231 1613 2311 4133 11071
47 109 191 349 911 3469
31 41 79 281 1279
5 19 101 499
7 41 199
17 79
31
Discoverer : Jens Kruse Andersen, 2002/10/06 (verified minimal, J. Brennen)
Sum : 206282
1087 1181 1543 2309 3727 6701 19543 93077
47 181 383 709 1487 6421 36767
67 101 163 389 2467 15173
17 31 113 1039 6353
7 41 463 2657
17 211 1097
97 443
173
Discoverer : Jens Kruse Andersen, 2002/10/06 (verified minimal, J. Brennen)
Sum : 1310077
10193 11839 14657 19231 26993 40639 75617 184351 533009
823 1409 2287 3881 6823 17489 54367 174329
293 439 797 1471 5333 18439 59981
73 179 337 1931 6553 20771
53 79 797 2311 7109
13 359 757 2399
173 199 821
13 311
149
Discoverer : Jens Kruse Andersen, 2002/10/07 (verified minimal by J.Brennen)
Sum : 15500585
255137 256651 259697 265819 276833 305611 392177 711259 1884833 6212491
757 1523 3061 5507 14389 43283 159541 586787 2163829
383 769 1223 4441 14447 58129 213623 788521
193 227 1609 5003 21841 77747 287449
17 691 1697 8419 27953 104851
337 503 3361 9767 38449
83 1429 3203 14341
673 887 5569
107 2341
1117
(note - no repetitions!)
Discoverer : Jack Brennen, 2002/10/07
Sum : 247033896
137639 767857 1423511 2220193 3306599 4900657 7340951 11493793 20001959 40016497 91381271
315109 327827 398341 543203 797029 1220147 2076421 4254083 10007269 25682387
6359 35257 72431 126913 211559 428137 1088831 2876593 7837559
14449 18587 27241 42323 108289 330347 893881 2480483
2069 4327 7541 32983 111029 281767 793301
1129 1607 12721 39023 85369 255767
239 5557 13151 23173 85199
2659 3797 5011 31013
569 607 13001
19 6197
3089
(note - no repetitions!)
Discoverer : Jack Brennen, 2002/10/07
Your name here?
Sizes 1 to 3 are identical to the above
Sum : 2
2Discoverer : Aristophenes, Cowboy Neal, and my pet cat "fluffy".
Sum : 12
3 7 2Discoverer : Phil Carmody. 2002/10/04.
Sum : 71
11 17 31
3 7
2
Discoverer : Nathan Russel, 2002/10/04.
Sum : 488
17 43 89 211
13 23 61
5 19
7
Discoverer : Jack Brennen, 2002/10/04
Sum : 2175
103 137 223 401 823
17 43 89 211
13 23 61
5 19
7
Discoverer : Jack Brennen, 2002/10/04
Sum : 9069
59 181 347 661 1499 3733
61 83 157 419 1117
11 37 131 349
13 47 109
17 31
7
Discoverer : Jack Brennen, 2002/10/04
Sum : 52292
29 823 1949 3463 5501 8887 18749
397 563 757 1019 1693 4931
83 97 131 337 1619
7 17 103 641
5 43 269
19 113
47
Discoverer : Jens Kruse Andersen, 2002/10/07
Sum : 449567
16249 18287 22153 28031 36217 47279 69193 131711
1019 1933 2939 4093 5531 10957 31259
457 503 577 719 2713 10151
23 37 71 997 3719
7 17 463 1361
5 223 449
109 113
2
Discoverer : Jens Kruse Andersen, 2002/10/07
Sum : 2271947
27697 30671 36097 44879 59473 84431 131617 236879 896113
1487 2713 4391 7297 12479 23593 52631 329617
613 839 1453 2591 5557 14519 138493
113 307 569 1483 4481 61987
97 131 457 1499 28753
17 163 521 13627
73 179 6553
53 3187
1567
Discoverer : Jack Brennen, 2002/10/07
Sum : 15500585
255137 256651 259697 265819 276833 305611 392177 711259 1884833 6212491
757 1523 3061 5507 14389 43283 159541 586787 2163829
383 769 1223 4441 14447 58129 213623 788521
193 227 1609 5003 21841 77747 287449
17 691 1697 8419 27953 104851
337 503 3361 9767 38449
83 1429 3203 14341
673 887 5569
107 2341
1117
Discoverer : Jack Brennen, 2002/10/07
Sum : 247033896
137639 767857 1423511 2220193 3306599 4900657 7340951 11493793 20001959 40016497 91381271
315109 327827 398341 543203 797029 1220147 2076421 4254083 10007269 25682387
6359 35257 72431 126913 211559 428137 1088831 2876593 7837559
14449 18587 27241 42323 108289 330347 893881 2480483
2069 4327 7541 32983 111029 281767 793301
1129 1607 12721 39023 85369 255767
239 5557 13151 23173 85199
2659 3797 5011 31013
569 607 13001
19 6197
3089
Discoverer : Jack Brennen, 2002/10/07
Your name here?
Fashion is a form of ugliness so intollerable that we have to alter it every six months. (Oscar Wilde)
Here are some logs:
# dups nodups 1 .7 2 2.5 3 4.3 4 5.7 6.2 5 6.9 7.7 6 8.7 9.1 7 10.3 10.9 8 12.2 13.0 9 14.1 14.6 10 16.6 11 19.3
It's curious to notice that the 'no duplications' rule started off making no difference, for the trivial sizes, and then caused the two families to diverge hugely, but now they appear to have converged again at size 10. It's clear that the parimes become much more sparse as the triangle grows, and therefore the chances of being able to tread on your own toes decreases. However, my guess is that there will be a larger minimal one with duplicates, and that at that point the cost will be quite large.
Thanks to Gary Chaffey for spotting the complete balls-up I made of the size 3 case!
Thanks to Jens for spotting the complete pigs ear I made of Jack's size 9 records (having 7 rows should have been a give-away!
Thanks to mistermac (John McNamara) for suggesting some alternative names, including "Prime Reduction Puzzle", and some quite funny ones that deserved to be used in a better context.
[*][* only joking]
Another hastily written page written by
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