mersenneforum.org generalized minimal (probable) primes
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2017-05-20, 15:46   #12
sweety439

Nov 2016

2×3×5×79 Posts

Quote:
 Originally Posted by sweety439 These problems are to find a prime of the form (k*b^n+c)/gcd(k+c,b-1) with integer n>=1 for fixed integers k, b and c, k>=1, b>=2, gcd(k,c)=1 and gcd(b,c)=1. For some (k,b,c), there cannot be any prime because of covering set (e.g. (k,b,c) = (78557,2,1), (334,10,-1) or (84687,6,-1)) or full algebra factors (e.g. (k,b,c) = (9,4,-1), (2500,16,1) or (9,4,-25) (the case (9,4,-25) can produce prime only for n=1)) or partial algebra factors (e.g. (k,b,c) = (144,28,-1), (25,17,-9) or (1369,30,-1)). It is conjectured that for every (k,b,c) which cannot be proven that they do not have any prime, there are infinitely primes of the form (k*b^n+c)/gcd(k+c,b-1). (Notice the special case: (k,b,c) = (8,128,1), it cannot have any prime but have neither covering set nor algebra factors) However, there are many such cases even not have a single known prime, like (21181,2,1), (2293,2,-1), (4,53,1), (1,185,-1), (1,38,1), (269,10,1), (197,7,-1), (4105,17,-9), (16,21,335), (5,36,821), but not all case will produce a minimal prime to base b, e.g. the form (197*7^n-1)/2 is the form 200{3} in base 7, but since 2 is already prime, the smallest prime of this form (if exists) will not be a minimal prime in base 7. The c=1 and gcd(k+c,b-1)=1 case is the Sierpinski problem base b, and the c=-1 and gcd(k+c,b-1)=1 case is the Riesel problem base b.
gcd(k+c,b-1) is the largest number that divides k*b^n+c for all n.

Note: gcd(0, m) = m for all integer m, and gcd(1, m) = 1 for all integer m.

Last fiddled with by sweety439 on 2017-05-20 at 15:47

2017-05-20, 17:11   #13
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

100011101010002 Posts

Quote:
 Originally Posted by sweety439 Note: gcd(0, m) = m for all integer m, and gcd(1, m) = 1 for all integer m.
Thank you, Captain Obvious!
In other news today, light travels faster than sound, and a minute contains 60 seconds.

2017-05-21, 03:45   #14
Xyzzy

"Mike"
Aug 2002

170358 Posts

Quote:
 Originally Posted by Batalov …and a minute contains 60 seconds.
Is that always true?

https://en.wikipedia.org/wiki/Leap_second

2017-05-21, 05:15   #15
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

23·7·163 Posts

Quote:
 Originally Posted by Xyzzy Is that always true?
It is just as true as "gcd(0, m) = m for all integer m".
For m=0, gcd(0, 0) = 24. 24 is a greatest common divisor of 0 and 0, because it divides both 0 and 0, and there is no higher number: see goo.gl/ASN4Ov ... and 24 ≠ 0

2017-05-21, 17:43   #16
sweety439

Nov 2016

2×3×5×79 Posts

Quote:
 Originally Posted by Batalov It is just as true as "gcd(0, m) = m for all integer m". For m=0, gcd(0, 0) = 24. 24 is a greatest common divisor of 0 and 0, because it divides both 0 and 0, and there is no higher number: see goo.gl/ASN4Ov ... and 24 ≠ 0
No, gcd(0, m) = m is true only for positive integer m.

Last fiddled with by sweety439 on 2017-05-21 at 17:44

2017-05-21, 22:03   #17
CRGreathouse

Aug 2006

22×1,483 Posts

Quote:
 Originally Posted by sweety439 No, gcd(0, m) = m is true only for positive integer m.
gcd(0, 0) = 0 so the rule holds in all cases except when you want to take 24 as maximal instead of 0 for humor.

2017-05-22, 14:29   #18
Dr Sardonicus

Feb 2017
Nowhere

2·32·199 Posts

Quote:
 Originally Posted by Batalov light travels faster than sound
Depends on the medium. Through a vacuum, it certainly does. Also through the air we breathe. But it takes a long, long time for the EM energy produced in the solar core to make its way through the interior of the sun, and out as sunshine. Sound waves travel through the interior of the sun much more quickly.

Last fiddled with by Dr Sardonicus on 2017-05-22 at 14:30

2017-05-22, 18:00   #19
sweety439

Nov 2016

2×3×5×79 Posts

Quote:
 Originally Posted by Dr Sardonicus Depends on the medium. Through a vacuum, it certainly does. Also through the air we breathe. But it takes a long, long time for the EM energy produced in the solar core to make its way through the interior of the sun, and out as sunshine. Sound waves travel through the interior of the sun much more quickly.
If the medium is not transparent, the the speed of (visible light) is zero, thus it is lower then that of sound. Besides, if the medium is vacuum, then the speed of sound is zero, since sound needs medium to spread.

2017-05-22, 18:05   #20
sweety439

Nov 2016

2×3×5×79 Posts

Quote:
 Originally Posted by Batalov Thank you, Captain Obvious! In other news today, light travels faster than sound, and a minute contains 60 seconds.
A minute does not always contain 60 seconds, since the definition of second is from the cesium atomic, it is not always 1/60 minute = 1/86400 day, since the definition of day is form earth. Besides, in Alaska and in Amazon forest, the length of "second" is not the same, since the distance of them to geocentric is different XDDD...

Last fiddled with by sweety439 on 2017-05-22 at 18:06

 2019-11-27, 09:36 #21 sweety439     Nov 2016 2×3×5×79 Posts Base 36 has only two unsolved family: (4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3
2019-11-27, 09:39   #22
sweety439

Nov 2016

1001010000102 Posts

Quote:
 Originally Posted by sweety439 Base 36 has only two unsolved family: (4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3
A (probable) prime was found:

(13998*40^12381+29)/13

Written in base 40, this number is Qa{U12380}X

This number is likely the second-largest "base 40 minimal prime"

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