44 Jean-Marie De Koninck

10k qk = pr pr+1

10 139 149

20 887 907

30 4 297 4 327

40 19 333 19 373

50 31 907 31 957

60 43 331 43 391

70 173 359 173 429

80 542 603 542 683

90 404 851 404 941

100 396 733 396 833

110 1 468 277 1 468 387

10k qk = pr pr+1

120 1 895 359 1 895 479

130 5 518 687 5 518 817

140 7 621 259 7 621 399

150 13 626 257 13 626 407

160 33 803 689 33 803 849

170 27 915 737 27 915 907

180 17 051 707 17 051 887

190 142 414 669 142 414 859

200 378 043 979 378 044 179

210 20 831 323 20 831 533

220 47 326 693 47 326 913

10k qk = pr pr+1

230 607 010 093 607 010 323

240 391 995 431 391 995 671

250 387 096 133 387 096 383

260 944 192 807 944 193 067

270 1 391 048 047 1 391 048 317

280 1 855 047 163 1 855 047 443

290 1 948 819 133 1 948 819 423

300 4 758 958 741 4 758 959 041

310 4 024 713 661 40 247 113 971

320 2 300 942 549 2 300 942 869

330 6 291 356 009 6 291 356 339

as for the smallest prime number pr such that pr+1 −pr = 1000, see the number

22 439 962 446 379 651;

• the number of digits in the fourth prime number n whose digits are 1 and 2

in alternation, that is of the form n = 1212 . . . 121; the prime numbers of this

form, with k digits, k 2000, are those with k = 7, 11, 43, 139, 627, 1 399,

1 597 and 1 979 digits respectively;

• the largest known prime p such that

3p

+2 is also prime; the other known prime

numbers p for which

3p

+ 2 is prime are 2 and 3.

140

• the only number n 2 such that

n2

=

m

3

for a certain number m, here with

m = 50; it is interesting to observe that K. Gyory [105] has established that

the only solution of n =

m

k

other than the ones with k = = 2 is the one

with n = 140, = 2, m = 50 and k = 3;

• the smallest number 1 which is not perfect or multi-perfect (we say that a

number n is multi-perfect if σ(n)/n is an integer) but whose harmonic mean is