Problems & Puzzles: Puzzles

Puzzle 531. Two products of consecutive primes a power JM Bergot poses the following puzzle: One notices that 2*3*5*7 + 37*41 = 1727, just one away from the cube of 12, 1728. Q. Can the sum of two products of consecutive primes equal a power?

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Contributions came from Jim Howell & J-C Colin.

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Jim wrote:

I have found the following solutions to puzzle 531.

2*3 + 2*3*5*7 = 216 = 6^3
3*5 + 2*3*5*7 = 225 = 15^2
5*7 + 13*17 = 256 = 2^8
7*11 + 17*19 = 400 = 20^2
11*13 + 67*71 = 4900 = 70^2

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Colin wrote:

The number k = p(i)*p(i+1) + p(i+2)*p(i+3) with i up to 9,999,997, is a (2 to 10) power only for :

i = 18242 : 203233*203249 + 203279*203293 = 82632101764 = 287458^2

The number k = p(i)*p(i+1) + p(i+3)*p(i+4) with i up to 9,999,996, is a (2 to 10) power only for :

i = 3 : 5*7 + 13*17 = 256 = 16^2 = 4^4 = 2^8
i = 4 : 7*11 + 17*19 = 400 = 20 ^2
i = 21 : 73*79 + 89*97 = 14400 = 120 ^2
i = 15569 : .170767*170773 + 170801*170809 = 58336740900 = 241530^2


The k = p(i)*p(i+1) + p(i+4)*p(i+5) can be a square :
i = 1257742 : 19784717*19784731 + 19784759*19784771 = 782872229861316 = 27979854^2

The number k = p(i)*p(i+1)*p(i+2) + p(i+3)*p(i+4) is a (2 to 10) power, for example, for:

i = 38 : 163*167*173+179*181 = 4741632 =168^3

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Jim Howell wrote again:

Here are some more solutions for puzzle 531:

3*5 + 5*7*11 = 400 = 20^2
17*19 + 19*23*29 = 12996 = 114^2
19*23*29 + 107*109 = 24336 = 156^2
23*29*31 + 251*257 = 85184 = 44^3

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Emmanuel Vantieghem wrote:

Here are some solutions to the puzzle 531 with p(j) < 1000, m < 10000.

Solutions of type p(j)p(j+1)+p(m)p(m+1) = Square (p(j)<1000,m<10000) :
5 * 7 + 13 * 17 = 256 = 16 ²
7 * 11 + 17 * 19 = 400 = 20 ²
11 * 13 + 67 * 71 = 4900 = 70 ²
19 * 23 + 107 * 109 = 12100 = 110 ²
37 * 41 + 373 * 379 = 142884 = 378 ²
41 * 43 + 877 * 881 = 774400 = 880 ²
73 * 79 + 89 * 97 = 14400 = 120 ²
79 * 83 + 137 * 139 = 25600 = 160 ²
79 * 83 + 1627 * 1637 = 2669956 = 1634 ²
83 * 89 + 401 * 409 = 171396 = 414 ²
97 * 101 + 337 * 347 = 126736 = 356 ²
101 * 103 + 389 * 397 = 164836 = 406 ²
101 * 103 + 5189 * 5197 = 26977636 = 5194 ²
127 * 131 + 4139 * 4153 = 17205904 = 4148 ²
127 * 131 + 4157 * 4159 = 17305600 = 4160 ²
163 * 167 + 6793 * 6803 = 46240000 = 6800 ²
181 * 191 + 3449 * 3457 = 11957764 = 3458 ²
191 * 193 + 2621 * 2633 = 6937956 = 2634 ²
211 * 223 + 3917 * 3919 = 15397776 = 3924 ²
229 * 233 + 13337 * 13339 = 177955600 = 13340 ²
233 * 239 + 9277 * 9281 = 86155524 = 9282 ²
241 * 251 + 479 * 487 = 293764 = 542 ²
241 * 251 + 1193 * 1201 = 1493284 = 1222 ²
263 * 269 + 359 * 367 = 202500 = 450 ²
271 * 277 + 12503 * 12511 = 156500100 = 12510 ²
277 * 281 + 641 * 643 = 490000 = 700 ²
277 * 281 + 2767 * 2777 = 7761796 = 2786 ²
277 * 281 + 19447 * 19457 = 378458116 = 19454 ²
283 * 293 + 349 * 353 = 206116 = 454 ²
283 * 293 + 8287 * 8291 = 68790436 = 8294 ²
293 * 307 + 44963 * 44971 = 2022121024 = 44968 ²
347 * 349 + 499 * 503 = 372100 = 610 ²
347 * 349 + 60527 * 60539 = 3664365156 = 60534 ²
349 * 353 + 30727 * 30757 = 945193536 = 30744 ²
353 * 359 + 479 * 487 = 360000 = 600 ²
367 * 373 + 22787 * 22807 = 519840000 = 22800 ²
373 * 379 + 23557 * 23561 = 555167844 = 23562 ²
389 * 397 + 431 * 433 = 341056 = 584 ²
433 * 439 + 10531 * 10559 = 111386916 = 10554 ²
439 * 443 + 733 * 739 = 736164 = 858 ²
443 * 449 + 743 * 751 = 756900 = 870 ²
457 * 461 + 1667 * 1669 = 2992900 = 1730 ²
461 * 463 + 2243 * 2251 = 5262436 = 2294 ²
467 * 479 + 919 * 929 = 1077444 = 1038 ²
479 * 487 + 709 * 719 = 743044 = 862 ²
487 * 491 + 8527 * 8537 = 73034116 = 8546 ²
499 * 503 + 1409 * 1423 = 2256004 = 1502 ²
541 * 547 + 929 * 937 = 1166400 = 1080 ²
593 * 599 + 877 * 881 = 1127844 = 1062 ²
613 * 617 + 94483 * 94513 = 8930250000 = 94500 ²
617 * 619 + 1201 * 1213 = 1838736 = 1356 ²
641 * 643 + 2269 * 2273 = 5569600 = 2360 ²
653 * 659 + 10223 * 10243 = 105144516 = 10254 ²
661 * 673 + 4091 * 4093 = 17189316 = 4146 ²
677 * 683 + 911 * 919 = 1299600 = 1140 ²
677 * 683 + 3041 * 3049 = 9734400 = 3120 ²
743 * 751 + 4951 * 4957 = 25100100 = 5010 ²
751 * 757 + 2137 * 2141 = 5143824 = 2268 ²
757 * 761 + 1741 * 1747 = 3617604 = 1902 ²
757 * 761 + 2381 * 2383 = 6250000 = 2500 ²
757 * 761 + 8447 * 8461 = 72046144 = 8488 ²
761 * 769 + 1217 * 1223 = 2073600 = 1440 ²
769 * 773 + 2017 * 2027 = 4682896 = 2164 ²
773 * 787 + 8669 * 8677 = 75829264 = 8708 ²
827 * 829 + 18013 * 18041 = 325658116 = 18046 ²
881 * 883 + 2213 * 2221 = 5692996 = 2386 ²
911 * 919 + 4133 * 4139 = 17943696 = 4236 ²
919 * 929 + 983 * 991 = 1827904 = 1352 ²
929 * 937 + 3691 * 3697 = 14516100 = 3810 ²
929 * 937 + 27191 * 27197 = 740384100 = 27210 ²
Solutions of type p(j)p(j+1) + p(m)p(m+1)p(m+2) = square :
2 * 3 + 2 * 3 * 5 = 36 = 6 ²
3 * 5 + 5 * 7 * 11 = 400 = 20 ²
11 * 13 + 71 * 73 * 79 = 409600 = 640 ²
17 * 19 + 19 * 23 * 29 = 12996 = 114 ²
67 * 71 + 659 * 661 * 673 = 293162884 = 17122 ²
101 * 103 + 733 * 739 * 743 = 402483844 = 20062 ²
107 * 109 + 19 * 23 * 29 = 24336 = 156 ²
191 * 193 + 73 * 79 * 83 = 515524 = 718 ²
521 * 523 + 19 * 23 * 29 = 285156 = 534 ²
839 * 853 + 89 * 97 * 101 = 1587600 = 1260 ²
911 * 919 + 1187 * 1193 * 1201 = 1701562500 = 41250 ²
Solutions of type p(j)p(j+1) + p(m)p(m+1)p(m+2)p(m+3) = square :
3 * 5 + 2 * 3 * 5 * 7 = 225 = 15 ²
337 * 347 + 7 * 11 * 13 * 17 = 133956 = 366 ²
Solutions of type p(j)p(j+1)p(j+2) + p(m)p(m+1)p(m+2) = square :
157 * 163 * 167 + 853 * 857 * 859 = 632220736 = 25144 ²
Solutions of type p(j)p(j+1)p(j+2) + p(m)p(m+1)p(m+2)p(m+3) = square :
919 * 929 * 937 + 97 * 101 * 103 * 107 = 907937424 = 30132 ²
I also found some cubes :
2293 * 2297 + 7283 * 7297 = 58411072 = 388 ³
3371 * 3373 + 3881 * 3889 = 26463592 = 298 ³
7417 * 7433 + 45077 * 45083 = 2087336952 = 1278 ³
179 * 181 + 163 * 167 * 173 = 4741632 = 168 ³
251 * 257 + 23 * 29 * 31 = 85184 = 44 ³
2 * 3 + 2 * 3 * 5 * 7 = 216 = 6 ³
And I am convinced that there are also higher powers : I do not see any reason why there wouldn't be !

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Giovanni Resta wrote (April 2010):

I played a little with Puzzle 531, using a straighforward
approach.
I simply generated, stored and sorted all the products of
consecutive primes up to 2^63 = 9,223,372,036,854,775,808
(this first step took about 25 seconds)
and then I used this table of about 146,000,000 numbers to
perform the searches.

In particular I searched for 3 subproblems:

1. HIGHER POWERS
Squares and cubes has been already found aplenty,
so I enumerated the higher powers up to 2^63,
apart from fourth powers, where I stopped earlier since there are too many.

4^4 = 5 x 7 + 13 x 17
50^4 = 757 x 761 + 2381 x 2383
90^4 = 89 x 97 x 101 + 8039 x 8053
638^4 = 225733 x 225749 + 338707 x 338717
948^4 = 72101 x 72103 + 895801 x 895813
1350^4 = 535991 x 535999 + 1741897 x 1741903
1392^4 = 730321 x 730339 + 1794757 x 1794761
.... too many solutions ...
54704^4 = 1376065433 x 1376065441 + 2657379421 x 2657379443
--------------------------
78^5 = 863 x 877 x 881 + 47119 x 47123
238^5 = 274867 x 274871 + 829501 x 829511
548^5 = 3956881 x 3956917 + 5810587 x 5810593
648^5 = 3392479 x 3392491 + 10136359 x 10136381
960^5 = 2900441 x 2900477 + 28407031 x 28407053
1158^5 = 912173 x 912187 + 45623069 x 45623093
1332^5 = 38940907 x 38940919 + 51735569 x 51735571
=> 2640^5 = 58176953 x 58176961 + 353347273 x 353347279
=> 2640^5 = 75589271 x 75589279 + 350035877 x 350035883
=> Note that 2640^5 can be expressed in two ways!
3192^5 = 375635411 x 375635431 + 436198307 x 436198313
4452^5 = 195300997 x 195301009 + 1307975917 x 1307975927
4728^5 = 512861159 x 512861203 + 1448985107 x 1448985113
5528^5 = 244138597 x 244138603 + 2258906777 x 2258906801
--------------------------
216^6 = 36^9 = 6^18 = 6586067 x 6586087 + 7627811 x 7627817
328^6 = 23620279 x 23620283 + 26216279 x 26216293
828^6 = 291635441 x 291635471 + 487022243 x 487022251
1046^6 = 666523843 x 666523853 + 930323101 x 930323117
--------------------------
100^7 = 10^14 = 5587837 x 5587847 + 8293121 x 8293141
224^7 = 58432687 x 58432711 + 157741373 x 157741379
--------------------------
no other higher powers apart from
2^8 = 5 x 7 + 13 x 17


2. ODD POWERS
Odd powers are difficult because one of the
addends must of the form 2 x 3 x 5 x ...
I found only two such powers, namely:
15^2 = 2 x 3 x 5 x 7 + 3 x 5
111546431^2 = 2 x 3 x ... x 23 + 111546427 x 111546433

3. AT LEAST 3 FACTORS
I searched for powers with at least 3 factors in each
addend. These are those below 2^63 and are all squares.

1328^2 = 47 x 53 x 59 + 5 x 7 x 11 x 13 x 17 x 19
3020^2 = 13 x 17 x 19 x 23 x 29 + 43 x 47 x 53 x 59
5690^2 = 97 x 101 x 103 + 23 x 29 x 31 x 37 x 41
25144^2 = 157 x 163 x 167 + 853 x 857 x 859
30132^2 = 97 x 101 x 103 x 107 + 919 x 929 x 937
10218408^2 = 43711 x 43717 x 43721 + 27527 x 27529 x 27539
15088776^2 = 46771 x 46807 x 46811 + 3331 x 3343 x 3347 x 3359
59056990^2 = 8447 x 8461 x 8467 + 1277 x 1279 x 1283 x 1289 x 1291
144655544^2 = 274837 x 274843 x 274847 + 54727 x 54751 x 54767
148036148^2 = 277003 x 277007 x 277021 + 86981 x 86993 x 87011
149019540^2 = 257687 x 257689 x 257707 + 172049 x 172069 x 172079
162956430^2 = 297589 x 297601 x 297607 + 58243 x 58271 x 58309
190970262^2 = 331579 x 331589 x 331603 + 21937 x 21943 x 21961
216202578^2 = 21961 x 21977 x 21991 + 14683 x 14699 x 14713 x 14717
290682116^2 = 420293 x 420307 x 420313 + 217199 x 217201 x 217207
509645000^2 = 610639 x 610651 x 610661 + 317563 x 317587 x 317591
927969308^2 = 868939 x 868943 x 868951 + 589639 x 589643 x 589681
1299155254^2 = 1012751 x 1012763 x 1012769 + 865801 x 865807 x 865817

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