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
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