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 2022-04-01, 22:17 #408 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 22×5×173 Posts This is the smallest base b (b>=2) such that there is a unique prime with period length n, but gcd(Phi(n,b),n) > 1, i.e. Phi(n,b) is not prime, but Phi(n,b)/gcd(Phi(n,b),n) is prime. If n is in https://oeis.org/A253235, then gcd(Phi(n,b),n) = 1 for all b, thus there cannot be such b, and if Schinzel's hypothesis H is true, then there are infinitely many such bases b for all n not in https://oeis.org/A253235 Attached Files
2022-04-07, 00:56   #409
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

22·5·173 Posts

List of primes of the form k*2^n+1 for -5000<=k<=5000 (k odd), -5000<=n<=5000 (n != 0)

we choose absolute value for negative numbers (i.e. k<0)
we choose numerator for noninteger rational numbers (i.e. n<0)
Attached Files
 Proth.txt (1,019.1 KB, 20 views)

2022-05-03, 08:09   #410
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

22·5·173 Posts

a-file for A039951, see https://oeis.org/A039951/a039951_1.txt
Attached Files
 A039951 a039951.txt (119.3 KB, 18 views)

2022-05-04, 11:40   #411
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1101100001002 Posts

Quote:
 Originally Posted by sweety439 List of primes of the form k*2^n+1 for -5000<=k<=5000 (k odd), -5000<=n<=5000 (n != 0) we choose absolute value for negative numbers (i.e. k<0) we choose numerator for noninteger rational numbers (i.e. n<0)
OEIS sequences for the data:

Smallest |n|:

A033809 (k>0, n>0)
A046067 (k>0, n>=0)
A078680 (k>0, n>0, allow even k)
A040076 (k>0, n>=0, allow even k)
A108129 (k<0, n>0)
A046069 (k<0, n>=0)
A050412 (k<0, n>0, allow even k)
A040081 (k<0, n>=0, allow even k)
A067760 (k>0, n<0)
A252168 (k<0, n<0)
A096502 (k<0, n<0, 2^|n| > |k|)
A276417 (k<0, n<0, 2^|n| < |k|)

A078683 (k>0, n>0, corresponding primes)
A057025 (k>0, n>=0, corresponding primes)
A050921 (k>0, n>=0, allow even k, corresponding primes)
A052333 (k<0, n>0, corresponding primes)
A057026 (k<0, n>=0, corresponding primes)
A038699 (k<0, n>=0, allow even k, corresponding primes)
A123252 (k>0, n<0, corresponding primes)
A096822 (k<0, n<0, 2^|n| > |k|, corresponding primes)

Last fiddled with by sweety439 on 2022-05-04 at 12:57

 2022-05-04, 11:53 #412 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 22×5×173 Posts Also the n-values for specific k: k=-15: A002237 (positive n), A059612 (negative n) k=-13: A001773 (positive n), A096818 (negative n) k=-11: A001772 (positive n), A096817 (negative n) k=-9: A002236 (positive n), A059610 (negative n) k=-7: A001771 (positive n), A059609 (negative n) k=-5: A001770 (positive n), A059608 (negative n) k=-3: A002235 (positive n), A050414 (negative n) k=-1: A000043 (positive n), A000043 (negative n) k=3: A002253 (positive n), A057732 (negative n) k=5: A002254 (positive n), A059242 (negative n) k=7: A032353 (positive n), A057195 (negative n) k=9: A002256 (positive n), A057196 (negative n) k=11: A002261 (positive n), A102633 (negative n) k=13: A032356 (positive n), A102634 (negative n) k=15: A002258 (positive n), A057197 (negative n)
 2022-05-04, 12:57 #413 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 22·5·173 Posts This is the dual problem of problem 49 of twin primes (see https://mersenneforum.org/forumdisplay.php?f=86, http://mersenneforum.org/showthread.php?t=10754, http://mersenneforum.org/showthread.php?t=6545) For odd k < 237 which are divisible by 3, does there always exist n>1 such that 2^n+k and 2^n-k are both primes? (for k = 237, such prime pairs cannot exist, since the same as twin primes of the form k*2^n+1 and k*2^n-1) status: Code: 3,3 9,5 15,5 21,5 27,5 33,6 39,7 45,6 51,9 57,8 63,unknown 69,7 75,8 81,9 87,unknown 93,8 99,7 105,7 111,7 117,8 123,7 129,9 135,14 141,unknown 147,10 153,8 159,11 165,9 171,unknown 177,8 183,8 189,15 195,unknown 201,unknown 207,unknown 213,10 219,unknown 225,11 231,9 remaining k values are {63, 87, 141, 171, 195, 201, 207, 219}, compare with the original twin prime conjecture for k*2^n+1 and k*2^n-1, its remaining k values are {111, 123, 153, 159, 171, 183, 189, 219, 225}, the only two mixed-remaining k values are 171 and 219
 2022-05-17, 19:38 #414 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 D8416 Posts OEIS sequences of smallest Proth primes (k*b^n+1), smallest Riesel primes (k*b^n-1), smallest dual Proth primes (b^n+k), smallest dual Riesel primes (b^n-k): Code: k smallest Proth primes (k*b^n+1) smallest Riesel primes (k*b^n-1) smallest dual Proth primes (b^n+k) smallest dual Riesel primes (b^n-k) 2 A119624 A119591 A138066 A255707 b-1 A305531 A?????? A076845 A113516 b+1 A?????? A?????? A346149 A178250 n A240234 A240235 A093324 A084746 (all entries in the list currently with no OEIS sequences (A??????) are Williams primes ((b+-1)*b^n+-1), for the data, see https://www.rieselprime.de/ziki/Williams_prime and http://harvey563.tripod.com/wills.txt, and there are also subsequences with prime indices in OEIS: A087139 and A122396) Last fiddled with by sweety439 on 2022-05-17 at 19:40
2022-05-21, 23:14   #415
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

D8416 Posts

Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal)

Next term (n=X1X1) is very large, such number
Attached Files
 palindromic mult.txt (169.4 KB, 7 views)

2022-05-21, 23:38   #416
chalsall
If I May

"Chris Halsall"
Sep 2002

53×199 Posts

Quote:
 Originally Posted by sweety439 Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal)
When you write "dozenal", is this simply base 12? I seem to remember seeing that word before...

As far as I understand it, pure math is the same no matter what base (let alone dimensions). More than happy to be corrected on that.

Programmers usually actually work in base two (even if they don't realize it).

2022-05-22, 02:57   #417
Dr Sardonicus

Feb 2017
Nowhere

73×17 Posts

Quote:
 Originally Posted by sweety439 Smallest k such that n*k is palindromic in dozenal (all n and k are written in dozenal) Next term (n=X1X1) is very large, such number
Assuming that n = X1X1 is a positive integer congruent to 1 (mod 12), then

$12^{\varphi(n)}\;-\;1$

is a base twelve palindromic number divisible by n. It may not be the smallest such.

Nil sapientiae odiosius obscuritate nimia

2022-05-22, 08:51   #418
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

22×5×173 Posts

Quote:
 Originally Posted by Dr Sardonicus Assuming that n = X1X1 is a positive integer congruent to 1 (mod 12), then $12^{\varphi(n)}\;-\;1$ is a base twelve palindromic number divisible by n. It may not be the smallest such. Nil sapientiae odiosius obscuritate nimia
No, X1X1 (base 12) = 17545 (base 10), it is hard to find a positive multiple of 17545 which is palindromic number in base 12 (A029957)

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