Cunningham numbers

A number

is a Cunningham number if it can be written as $C^{+}(b,k)=b^k+1$ or $C^{-}(b,k)=b^k-1$ for $b,k

1$">.

For example, $282475250$ is a Cunningham number because it is equal to $7^{10}+1$ .

For any fixed base $b$ , the exponents $k$ for which $C^{+}(b,k)$ or $C^{-}(b,k)$ is prime are in general very scarce.

This is due to the fact that $b^k-1$ is always divisible by $b-1$ , and thus it can be prime only if $b=2$ . Moreover $b^{h\cdot k}-1$ is always divisible by $b^h-1$ , thus a necessary condition for $2^k-1$ to be prime is that $k$ is prime as well.

On the other side, $b^k+1$ is always divisible by 2 if $b$ is odd, and by $b+1$ , if $k$ is odd. If $k$ is even because it is of the form $k=2^h\cdot q$ with 1$"> odd, then $b^k+1$ is divisible by $b^{2^h}+1$ , hence the only candidates left for primality are of the form $C^{+}(b,2^h)$ with $b$ even.

In general, the factorization of Cunningham numbers with small bases (and large exponents) has been and is a popular topic in (computational) number theory.

The first Cunningham numbers are 3, 5, 7, 8, 9, 10, 15, 17, 24, 26, 28, 31, 33, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Cunningham numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Cunningham Number
Wikipedia, Cunningham number
OEIS, Sequence A080262

Cunningham numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 101 8101 22501 + 9926136901 ABA 24 50 242 + 2584123765442 aban 10 15 17 + 999998000002 abundant 24 48 80 + 49999040 Achilles 288 675 9800 + 15061377048200 admirable 24 120 224 + 86769224 alt.fact. 101 alternating 10 50 63 + 987090725 amenable 17 24 28 + 999950885 amicable 356408 666094293315 anti-perfect 244 apocalyptic 224 226 528 + 29930 arithmetic 15 17 31 + 9999999 astonishing 15 balanced p. 257 13457 30977 + 9748402757 Bell 15 bemirp 1601 betrothed 48 195 2024 + 26409320 binomial 10 15 28 + 6294047334435 brilliant 10 15 35 + 998307217 c.decagonal 31 101 32401 + 348305329443601 c.heptagonal 197 7568 176401 + 127740041301797 c.nonagonal 10 28 325 + 575598885856165 c.pentagonal 31 226 324901 + 974193625805626 c.square 145 4901 166465 + 255821727047185 c.triangular 10 31 901 + 796926124814401 cake 15 26 310248 + 3940224 Carmichael 1729 46657 2433601 + 432081216001 Chen 17 31 37 + 99720197 compositorial 24 congruent 15 24 28 + 9999999 constructible 10 15 17 + 17180131328 Cullen 65 2049 1048577 + 2473901162497 Curzon 26 33 50 + 199883045 cyclic 15 17 31 + 9998243 D-number 15 33 63 + 7011903 d-powerful 24 63 224 + 9865882 de Polignac 127 3845 10001 + 99800099 decagonal 10 126 2047 deceptive 1729 7777 10001 + 54721373477 deficient 10 15 17 + 9999999 dig.balanced 10 15 35 + 199967882 double fact. 15 48 droll 4224 Duffinian 35 50 63 + 9998243 eban 50 32040 32042 + 32042064046042 economical 10 15 17 + 19980901 emirp 17 31 37 + 199374401 emirpimes 15 26 122 + 99920017 equidigital 10 15 17 + 19980901 eRAP 24 170 23408 + 991645522595 esthetic 10 65 101 + 123456765432101 Eulerian 26 120 evil 10 15 17 + 1000000001 factorial 24 120 5040 fibodiv 28 122 244 + 17690 Friedman 126 127 2501 + 978122 frugal 33125 57122 101125 + 910953125 gapful 120 143 170 + 99992558655 Gilda 440 good prime 17 37 101 + 139240001 happy 10 28 31 + 9991920 harmonic 28 32760 Harshad 10 24 48 + 9998200080 heptagonal 342 783 143640 + 1324147312657 hex 37 127 217 + 9994070595601 hexagonal 15 28 120 + 575598885856165 highly composite 24 48 120 + 17297280 hoax 483 17575 24963 + 99820080 Hogben 31 8191 Honaker 3137 4357 13457 + 960752017 hungry 17 hyperperfect 28 325 2924101 2082096901 iban 10 17 24 + 774401 iccanobiF 124 idoneal 10 15 24 + 1848 inconsummate 63 65 195 + 994008 interprime 15 26 50 + 99960005 Jordan-Polya 24 48 120 + 25920 junction 101 511 513 + 99897345 Kaprekar 99 999 7777 + 99999999999999 katadrome 10 31 50 + 9410 Lehmer 15 255 511 + 945192284101 Leyland 17 145 lonely 120 lucky 15 31 33 + 9884737 Lynch-Bell 15 24 48 + 1923768 magic 15 65 73599240 magnanimous 50 65 101 + 40001 metadrome 15 17 24 + 13457 modest 26 399 511 + 1968075768 Moran 63 195 399 + 99920017 Motzkin 127 323 2188 nialpdrome 10 31 33 + 99999999999999 nonagonal 24 325 22200 + 211162109090725 nude 15 24 33 + 496621224 oban 10 15 17 + 999 octagonal 65 1680 12545 + 461777249934008 odious 26 28 31 + 999999999 Ormiston 215297 820837 876097 + 1980784037 palindromic 33 99 101 + 995570353075599 palprime 101 pancake 37 1226 41617 + 63955431761797 panconsummate 10 15 24 + 257 pandigital 15 99 120 + 9743861520 partition 15 101 14883 pentagonal 35 145 1001 + 127942295300965 perfect 28 pernicious 10 17 24 + 9991922 Perrin 10 17 3480 persistent 1608972543 1635798024 3584297160 + 99241380675 Pierpont 17 37 257 + 142657607172097 plaindrome 15 17 24 + 899999999999999 Poulet 1729 2047 46657 + 598865079758401 powerful 288 675 9800 + 511643454094368 practical 24 28 48 + 9991920 prim.abundant 2210 3126 4095 + 95942024 prime 17 31 37 + 999920001601 primeval 37 pronic 342 1332 Proth 17 33 65 + 998848331777 pseudoperfect 24 28 48 + 999999 rare 65 repdigit 33 99 999 + 99999999999999 repfigit 28 197 2208 repunit 15 31 63 + 562949953421311 Rhonda 74511423 151225321128 Ruth-Aaron 15 24 50 + 996383276099 Saint-Exupery 6240 192720 49970760 + 205951200 Sastry 528 13224 self 31 143 244 + 999761162 semiprime 10 15 26 + 99960005 sliding 65 101 290 + 100000000000001 Smith 483 728 1935 + 99960003 Sophie Germain 1601 12101 25601 + 9960040001 sphenic 170 195 255 + 99980002 star 37 3601 352837 + 32530984516837 straight-line 840 999 7777 + 99999999999999 strobogrammatic 101 1001 10001 + 100018000810001 strong prime 17 37 101 + 99800101 super Niven 10 24 48 + 48400440000 super-d 31 127 511 + 9966650 superabundant 24 48 120 + 5040 tau 24 80 288 + 999128880 taxicab 1729 1092728 3375001 + 729729243027001 tetrahedral 10 35 120 + 435730689800 tetranacci 15 401 triangular 10 15 28 + 937385441796000 tribonacci 24 trimorphic 24 99 624 + 99999999999999 truncatable prime 17 31 37 + 24337 twin 17 31 101 + 995402501 uban 10 15 17 + 98010099000026 Ulam 26 28 48 + 9999999 undulating 101 242 323 + 232323 unprimeable 325 530 624 + 9991922 untouchable 120 124 288 + 998000 upside-down 28 37 82 + 241142869968 vampire 16265088 1052158968 1328675400 + 6769833840 wasteful 24 26 28 + 9999999 weak prime 31 401 577 + 99720197 weird 357910 Woodall 17 63 80 + 531726889113615 Zeisel 1729 Zuckerman 15 24 224 + 1212223488 Zumkeller 24 28 48 + 97968 zygodrome 33 99 999 + 557755556688800