- It is well known that for the expression

(n+1)^x - n^x

to yield a prime number, x must itself be prime.

(And if x is a prime p the same expression contains only prime

factors of the form p*2k + 1)

Here's a question: For every prime p, is there always an n such that

(n+1)^p - n^p is prime?

I think so. Here are the lowest 15 values of n which yield a prime

for the first 15 primes (p,n):

(2,1) (3,1) (5,1) (7,1) (11,5) (13,1) (17,1) (19,1) (23,5) (29,2)

(31,1) (37,39) (41,6) (43,4) (47,12)

(For example (12+1)^47 - 12^47 is prime. )

Here is a list (p,n) which shows only increasing n:

(2,1) (11,5) (37,39) (73,46) (137,327) (239,367) (307,402) (419,781)

(443,1774)

(For example (1774+1)^443 - 1774^443 is prime. (Over 1400 digits, my

biggest ever found , wheee! :) )

Another question is this: Given a prime p is there an upper bound for

n before which we can be sure to generate a prime?

The value (1 + 1/e) crops up alot in my dabblings, so for fun I'll

guess the following:

(n+1)^p - n^p is prime for some n < p^(1+1/e)

Oh the rigour of it all (not!)...

Mark - Hi all,

>

last year:-

> The 2 cases n=48 and n=61 are tough, and resisted a solution until

> 27 48^58543-47^58543 98425 Jean-Louis Charton 11/2005

I'm currently working on 10^p-9^p and 138^p-137^p. 138^p-137^p

> 28 61^54517-60^54517 97331 Jean-Louis Charton 09/2005

>

appears to be harder than 48 and 61.

Some time ago, I also started to build a table of the first n such

that (n+1)^p-n^p is prime or at least PRP. Here is the result :

p n n/p

2 1 0,50

3 1 0,33

5 1 0,20

7 1 0,14

11 5 0,45

13 1 0,07

17 1 0,05

19 1 0,05

23 5 0,21

29 2 0,06

31 1 0,03

37 39 1,05

41 6 0,14

43 4 0,09

47 12 0,25

53 2 0,03

59 2 0,03

61 1 0,01

67 6 0,08

71 17 0,23

73 46 0,63

79 7 0,08

83 5 0,06

89 1 0,01

97 25 0,25

101 2 0,01

103 41 0,39

107 1 0.00

109 12 0,11

113 7 0,06

127 1 0.00

131 7 0,05

137 327 2,38

139 7 0,05

149 8 0,05

151 44 0,29

157 26 0,16

163 12 0,07

167 75 0,44

173 14 0,08

179 51 0,28

181 110 0,60

191 4 0,02

193 14 0,07

197 49 0,24

199 286 1,43

211 15 0,07

223 4 0,01

227 39 0,17

229 22 0,09

233 109 0,46

239 367 1,53

241 22 0,09

251 67 0,26

257 27 0,10

263 95 0,36

269 80 0,29

271 149 0,54

277 2 0.00

281 142 0,50

283 3 0,01

293 11 0,03

307 402 1,30

311 3 0.00

313 44 0,14

317 10 0,03

331 82 0,24

337 20 0,05

347 95 0,27

349 4 0,01

353 108 0,30

359 349 0,97

367 127 0,34

373 303 0,81

379 37 0,09

383 3 0.00

389 162 0,41

397 119 0,29

401 8 0,01

409 128 0,31

419 781 1,86

421 5 0,01

431 25 0,05

433 75 0,17

439 189 0,43

443 1774 4,00

449 197 0,43

457 360 0,78

461 13 0,02

463 128 0,27

467 155 0,33

479 20 0,04

487 210 0,43

491 1550 3,15

499 3 0.00

503 1105 2,19

509 389 0,76

521 1 0.00

523 16 0,03

541 3 0.00

547 32 0,05

557 353 0,63

563 4 0.00

569 164 0,28

571 16 0,02

577 207 0,35

587 154 0,26

593 2175 3,66

599 3 0.00

601 604 1,00

607 1 0.00

613 104 0,16

617 348 0,56

619 251 0,40

631 128 0,20

641 1469 2,29

643 12 0,01

647 2 0.00

653 1167 1,78

659 13 0,01

661 100 0,15

673 22 0,03

677 193 0,28

683 1147 1,67

691 46 0,06

701 54 0,07

709 4 0.00

719 252 0,35

727 209 0,28

733 822 1,12

739 165 0,22

743 4 0.00

751 429 0,57

757 572 0,75

761 1973 2,59

769 56 0,07

773 27 0,03

787 81 0,10

797 33 0,04

809 509 0,62

811 73 0,09

821 180 0,21

823 94 0,11

827 91 0,11

829 101 0,12

839 359 0,42

853 29 0,03

857 81 0,09

859 155 0,18

863 629 0,72

877 21 0,02

881 666 0,75

883 154 0,17

887 329 0,37

907 172 0,18

911 722 0,79

919 943 1,02

929 264 0,28

937 459 0,48

941 989 1,05

947 699 0,73

953 968 1,01

967 1442 1,49

971 950 0,97

977 163 0,16

983 353 0,35

991 653 0,65

997 88 0,08

1009 553 0,54

1013 99 0,09

1019 3579 3,51

1021 889 0,87

1031 1654 1,60

1033 923 0,89

1039 5 0.00

1049 2741 2,61

1051 40 0,03

1061 2 0.00

1063 274 0,25

1069 339 0,31

1087 45 0,04

1091 72 0,06

1093 416 0,38

1097 1733 1,57

1103 347 0,31

1109 80 0,07

1117 90 0,08

1123 58 0,05

1129 10 0.00

1151 1094 0,95

1153 488 0,42

1163 39 0,03

1171 135 0,11

1181 33 0,02

1187 50 0,04

1193 3 0.00

1201 290 0,24

1213 1315 1,08

1217 473 0,38

1223 2790 2,28

1229 2122 1,72

1231 349 0,28

1237 414 0,33

1249 593 0,47

1259 424 0,33

1277 44 0,03

1279 1 0.00

1283 2058 1,60

1289 5143 3,98

1291 384 0,29

1297 871 0,67

1301 69 0,05

1303 55 0,04

1307 96 0,07

1319 490 0,37

1321 630 0,47

1327 6 0.00

1361 874 0,64

In fact, it seems that most of the time the first n is small and even

<= p but there are a few counterexamples.

My actual record is for p=443 for which n = 1774 (n/p = 4.0045).

I believe that the 2 following assertions are both true even if (of

course) they seems currently impossible to demonstrate:

- For every prime p there exist an integer n such that (n+1)^p-n^p is

prime (1)

- For every integer n >= 1 there exist a prime p such that (n+1)^p-

n^p is prime (2)

And more than that, I believe that the sets of such n (1) and such p

(2) are infinite

J-L