Problems & Puzzles: Puzzles

Puzzle 1021. p(k)+p(k+1)+1 Paolo Lava proposes the following nice puzzle: Least prime p(k) such that p(k) + p(k+1) + 1 is squarefree and with n prime factors 1 5 -> 5 + 7 + 1 = 13. 2 2 -> 2 + 3 + 1 = 6 = 2*3. 3 191 -> 191 + 193 + 1 = 5*7*11. 4 1021 -> 1201 + 1213 + 1 = 2415 = 3*5*7*23. 5 20521 -> 20521 + 20533 +1 = 41055 = 3*5*7*17*23. 6 390001 -> 390001 + 390043 + 1 = 780045 = 3*5*7*17*19*23. and Least prime p(k) such that p(k) + p(k+1) + p(k+2) is squarefree and with n prime factors 1 5 -> 5 + 7 + 11 = 23. 2 2 -> 2 + 3 + 5 = 10 = 2*5. 3 331 -> 331 + 337 + 347 = 1015 = 5*7*29. 4 3049 -> 3049 + 3061 + 3067 = 9177 = 3*7*19*23. 5 35227 -> 35227 + 35251 + 35257 = 105735 = 3*5*7*19*53. 6 299903 -> 299903 + 299909 + 299933 = 899745 = 3*5*7*11*19*41. Q. Find terms for n>6. Is there a demonstration that such numbers exist for any n?

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On the week 24-30 October 2020 contributions came from V. F. Izquierdo, Giorgos Kalogeropoulos, PaulCleary, Oscar Volpatti, Emmanuel Vantieghem.

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Vicente wrote:

For prime[k]+prime[k+1]+1:

7, 2.424.937, 3*5*7*11*13*17*19

8, 205.307.611, 3*5*7*11*13*23*29*41

9, 4.071.444.907, 3*5*7*11*13*17*19*23*73

10, 74.345.698.897, 3*5*7*11*13*17*19*23*31*43

11, 5.444.525.719.111, 3*5*7*11*13*17*19*23*31*47*67

12, 87.193.673.083.987, 3*5*7*11*13*17*19*23*29*31*37*47

 

For prime[k]+prime[k+1]+prime[k+2]:
 

7, 7.234.517, 3*5*7*11*19*23*43

8, 158.055.193, 3*5*7*11*17*19*31*41

9, 1.524.467.891, 3*5*7*11*13*17*19*23*41

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Giorgos wrote:

I have found 100 terms for each case. See this file.

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Paul wrote:

I think these are the minimum prime p(k) for both types up to 20 primes each

 

7 -> 2424907 + 2424937 + 1 = 4849845 = 3 x 5 x 7 x 11 x 13 x 17 x 19

8 -> 158055193 + 158055199 + 158055223 = 474165615 = 3 x 5 x 7 x 11 x 17 x 19 x 31 x 41

9 -> 4071444847 + 4071444907 + 1 = 8142889755 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 73

10 -> 74345698897 + 74345698957 + 1 = 148691397855 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 31 x 43

11 -> 5444525719111 + 5444525719153 + 1 = 10889051438265 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 31 x 47 x 67

12 -> 87193673083987 + 87193673084047 + 1 = 174387346168035 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 47

13 -> 6428314689524191 + 6428314689524263 + 1 = 12856629379048455 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 41 x 53 x 59

14 -> 219136252305472951 + 219136252305473053 + 1 = 438272504610946005 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 67

15 -> 15506131856082359923 + 15506131856082360061 + 1 = 31012263712164719985 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 37 x 41 x 43 x 47 x 53 x 59

16 -> 1052297628357522954853 + 1052297628357522954961 + 1 = 2104595256715045909815 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 43 x 47 x 59 x 67 x 71

17 -> 48549698841393869136463 + 48549698841393869136571 + 1 = 97099397682787738273035 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 101

18 -> 2829235940486873030746081 + 2829235940486873030746183 + 1 = 5658471880973746061492265 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 67 x 71 x 73

19 -> 198422619164776743160946947 + 198422619164776743160947097 + 1 = 396845238329553486321894045 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 61 x 67 x 101

20 -> 19916760708314661900454948627 + 19916760708314661900454948777 + 1 = 39833521416629323800909897405 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 71 x 79 x 83 x 89

 

 

7 -> 7234517 + 7234541 + 7234547 = 21703605 = 3 x 5 x 7 x 11 x 19 x 23 x 43

8 -> 158055193 + 158055199 + 158055223 = 474165615 = 3 x 5 x 7 x 11 x 17 x 19 x 31 x 41

9 -> 1524467891 + 1524467963 + 1524467981 = 4573403835 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 41

10 -> 146981019161 + 146981019173 + 146981019221 = 440943057555 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 59 x 67

11 -> 2924264157767 + 2924264157791 + 2924264157887 = 8772792473445 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 31 x 43 x 59

12 -> 124113516642097 + 124113516642109 + 124113516642139 = 372340549926345 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 47 x 79

13 -> 4535946133123387 + 4535946133123399 + 4535946133123429 = 13607838399370215 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 37 x 41 x 47 x 59

14 -> 132476253972317237 + 132476253972317261 + 132476253972317267 = 397428761916951765 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 43 x 47 x 53

15 -> 7276195760630481613 + 7276195760630481709 + 7276195760630481733 = 21828587281891445055 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 71

16 -> 711930353432117784941 + 711930353432117784953 + 711930353432117785001 = 2135791060296353354895 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 47 x 59 x 61 x 83

17 -> 37511148993323851080139 + 37511148993323851080259 + 37511148993323851080307 = 112533446979971553240705 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 41 x 43 x 47 x 53 x 59 x 61 x 71

18 -> 3130872412489286082022393 + 3130872412489286082022399 + 3130872412489286082022423 = 9392617237467858246067215 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 47 x 53 x 59 x 61 x 71 x 97

19 -> 185807294787963924506776321 + 185807294787963924506776333 + 185807294787963924506776381 = 557421884363891773520329035 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 59 x 61 x 67 x 73 x 103

20 -> 11071065345213582903890991271 + 11071065345213582903890991301 + 11071065345213582903890991313 = 33213196035640748711672973885 = 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 61 x 67 x 79 x 107

 

I see no reason that there wouldn’t be a solution for any n.  All of the solutions above were the first of many solutions for each p(k), going forward, there’s bound to be a smallest solution for any p(k).

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Oscar wrote:

I solved puzzle 1021 by enumerating odd squarefree numbers x and checking if they satisfy the desired equalities.

 

Equality x = p(k)+p(k+1)+1.

Given m2 = (x-1)/2, I found the two primes closest to m2, with p1 <= m2 < p2;

the only candidate pair of consecutive primes is (p1,p2).

 

Equality x = p(k)+p(k+1)+p(k+2).

Given m3 = floor(x/3), I found the four primes closest to m3, with p0 < p1 <= m3 < p2 < p3;

the only candidate triplets of consecutive primes are (p0,p1,p2), and (p1,p2,p3).

 

I checked odd squarefree numbers with prime factors not exceeding p(41) = 179;

for 7 <= n <= 40, I stored the smallest solution x with n factors, along with the related first prime p(k). 

 

The smallest odd squarefree number with n factors and out of this search is:

y = p(2)*p(3)*...*p(n)*181.

If y < x, then p(k) may not be minimal, and is listed with a question mark.

 

 

Equality x = p(k)+p(k+1)+1.

n  p(k)

01 5

02 2

03 191

04 1201

05 20521

06 390001

07 2424907

08 205307593

09 4071444847

10 74345698897

11 5444525719111

12 87193673083987

13 6428314689524191

14 219136252305472951

15 15506131856082359923

16 1052297628357522954853

17 48549698841393869136463

18 2829235940486873030746081

19 198422619164776743160946947

20 19916760708314661900454948627

21 1223562158599081691927806997137

22 115091894953327929993618204795283

23 8935674444249093596148774500752333

24 788162890171289663958107736305281963

25 83965729674895371306294451998157417507

26 8252204134951996821797540860837187913457

27 994146448716049092572281397739957191191867

28 91562743803782414676909980818950030486403663

29 12434892628691661325342187578558874323763618731

30 1511446749406898849706820771504946983152446540221 ?

31 240033830267457311016625805430550282925011330314547 ?

32 26436509654512070726009420604773599825447945987648963 ?

33 2935948568431057519791375900106007606745885878546170057

34 537145492054436878640008552615886882534036758094445893737 ?

35 79725012068889856270623717911279673627377713402021296665743 ?

36 10660714038984307666941905656747223902832165975931362729610253 ?

37 1883469429517692238987645577235708103786868136484463268712109851 ?

38 452937492829625345846838408588164983731173084900094178054653099007 ?
 

Equality x = p(k)+p(k+1)+p(k+2).

n  p(k)

01 5

02 2

03 331

04 3049

05 35227

06 299903

07 7234517

08 158055193

09 1524467891

10 146981019161

11 2924264157767

12 124113516642097

13 4535946133123387

14 132476253972317237

15 7276195760630481613

16 711930353432117784941

17 37511148993323851080139

18 3130872412489286082022393

19 185807294787963924506776321

20 11071065345213582903890991271

21 911024385458544953432961429287

22 73360943039347367332627440456697

23 4916024930368658091118484684618267

24 541149395986761160884132521968599251

25 56716179621663208397233957806441154009

26 7426884816150875644452121061900231920621 ?

27 480644324429304014052020896687401734836669

28 76004541555243324243672181504799118603524003

29 6718344946510775967205173814257750896888656679

30 943419964107509612258113580435230501844386261691 ?

31 149081959366332492239352742202974859684585038244029 ?

32 21268456061231687301380236091777509157769070424044069 ?

33 2713738906345350356232937966320751370838501451156715087 ?

34 408410765776958712982475163385681271055072647607811545331 ?

35 62326987852417489554492402971408090173921186016927920175023 ?

36 10848429162469733103836683865434487827407493418190555919091773 ?

37 1482490453810903337676088090159839206199813568437188144846472571 ?

38 263363474625361792072248895728988305415059361078573828505874458453 ?

 

He sent attached files w2.txt and w3.txt contain all primes and sum factorizations, available on request.

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Emmanuel wrote:

Solutions first kind :
n         p(k)             Prime factors of  p(k)+p(k+1)+1
7   2424907           3, 5, 7, 11, 13, 17, 19
8   205307593       3, 5, 7, 11, 13, 23, 29, 41
9   4071444847     3, 5, 7, 11, 13, 17, 19, 23, 73

Solutions second kind :
n         p(k)             Prime factors of  p(k)+p(k+1)+p(k+2)
7   7234517           3, 5, 7, 11, 19, 23, 43
8   158055193       3, 5, 7, 11, 17, 19, 31, 41
9   1524467891     3, 5, 7, 11, 13, 17, 19, 23, 41

I cannot imagine any reason why there would not be a solution for every  n.

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Seiji Tomita wrote on November 14, 2020:

About Puzzle 1021.

Search range: p(k)<10^10

n=7
2424907 + 2424937 + 1 = 3*5*7*11*13*17*19

n=8
205307593 + 205307611 + 1 = 3*5*7*11*13*23*29*41

n=9
4071444847 + 4071444907 + 1 = 3*5*7*11*13*17*19*23*73


n=7
7234517 + 7234541 + 7234547 = 3*5*7*11*19*23*43

n=8
158055193 + 158055199 + 158055223 = 3*5*7*11*17*19*31*41

n=9
1524467891 + 1524467963 + 1524467981 = 3*5*7*11*13*17*19*23*41

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