Minimal set of the strings for primes with at least two digits

[sweety439]

All known minimal primes (start with b+1) in bases 2<=b<=16: (data for bases 2, 3, 4, 5, 6, 8, 10, 12 are known to be complete)

Also three unsolved families are known:

Base 11: 5777...777
Base 13: 9555...555
Base 16: DBBB...BBB

[Uncwilly]

Rather than spilling out each new thought that comes to your head when it does. Try writing them down off line and posting only 1 well formatted post every couple of days. Again think about formatting it nicely in your word processor and then making a PDF that you can later update.

I am tired of seeing you posting drival serval times a day.

[sweety439]

fixed typo in the file: 3*16^n+1 should be 3*4^n+1, 12^((n+1)/2) +/- 5 should be 12^((n+2)/2) +/- 5

[VBCurtis]

You have an "edit" button. Use it. ONE pdf for this crap, not a new post with a new PDF every time you change a sentence. Edit the post, edit the PDF

If you were limited to one post per day on the forum, would this be it?

I'm deleting your first PDF and its post.

[sweety439]

Originally Posted by sweety439

* The smallest Williams prime with 4th kind base b (for b != 1 mod 3): [not minimal prime (start with 2 digits) if either b is prime or base b has smaller generalized Fermat prime, but for the case that b is prime, it is still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

7, 13, 31, 43, 73, 811, 1453, 157, 211, 241, 307, 3768826516993, 421, 463, 12697, 601, 18253, 757, 615334471, 27901, 1107296257, 1123, 44101, 1726273, 1483, 2372761, 1723, 75853, 87121, 93151, 106033, 599298932737, 2551, 158981126352779044590102826209115342318059775372698133871491241388097301966680877821738760704616125782843491355455960710073030287313404870590681666644752545879191893959727029866211537628677981607279205572507381073830401006677162824033234341436459420880686565908174585159142942438136179315586329074318947952541865853, 151687, 2971, 178753, 3307, 3541, 1338153989063049216000000000000000000001, 3907, 48326086052867645032352571108528903615254734667108057821332757600957454538355546211631290156513879123036351230974951391062798157776810891656336682957284917485088940693788242992185798654992956966627018064055387274320725152943868432582696386314597516885379356294528772183874293272350708412107233383892387582454781698467578958840732553153

* The smallest prime of the form 2*b^n+1 (for b != 1 mod 3): [not minimal prime (start with 2 digits) if b<=2]

5, 7, 11, 13, 17, 19, 23, 3457, 29, 31, 13555929465559461990942712143872578804076607708197374744547, 37, 41, 43, 47, 1153, 53, 1459, 59, 61, 65537, 67, 71, 73, 2*38^2729+1, 79, 83, 3529, 89, 4051, 82823796591884729837907950243851987042491027688029791782033968173988787397927431168748344242980462637086843228831225333542602440512725127029105275975234384910715377295392116427292929375082823988662090607733781357479215392846048752706418227733688234263166843856633793191822664770551012658601887, 97, 101, 103, 107, 109, 113, 370387, 410759, 432001, 236522599840432068647134316649762315445236710001482847056204302486382634336257, 127

* The smallest prime of the form 2*b^n-1: [not minimal prime (start with 2 digits) if b<=2]

3, 5, 7, 1249, 11, 13, 127, 17, 19, 241, 23, 337, 76831, 29, 31, 577, 647, 37, 20479999999999, 41, 43, 296071777, 47, 1249, 617831551, 53, 1567, 15387133080032326246081223292828787411221911122916017220126284227825703776392672467768318856009763825207593900596158761682711294895921233392537083406917227083982402321012446032594528728383203531755841, 59, 61, 2147483647, 2582935937, 67, 3676531249, 71, 73, 2887, 3041, 79, 3361, 83, 3697, 7496191, 89, 4231, 9759361, 10616831, 97, 4999, 101, 103, 14762783749438524018088313240622157671545425891033638774020213131211643094561, 107, 109, 6271, 113, 390223, 6961, 1555199999, 453961, 7687, 7937, 127

* The smallest prime of the form b^n+2 (for b == 3, 5 mod 6): [not minimal prime (start with 2 digits) if either b<=2 or b is prime, but still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

5, 7, 11, 13, 17, 19, 23, 952809757913929, 29, 31, 1091, 37, 41, 43, 47, 885233716287722386108568808645559198522547790058305212262181780420828956357982973084581935827930464156048602918053397761948271781610736426217362565287242033121579185919812362859356307201329, 53, 1174711139839, 59, 61, 250049

* The smallest prime of the form b^n-2 (for odd b): [not minimal prime (start with 2 digits) if b<=2]

7, 3, 5, 7, 14639, 11, 13, 24137567, 17, 19, 480250763996501976790165756943039, 23, 727, 839, 29, 31, 1223, 1367, 37, 2825759, 41, 43, 2207, 47, 45767944570399, 7890479, 53, 1176246293903439667999, 12117359, 59, 61

* The smallest prime of the form 3*b^n+1 (for even b): [not minimal prime (start with 2 digits) if b<=3]

7, 13, 19, 193, 31, 37, 43, 769, 17497, 61, 67, 73, 79, 40478785537, 81001, 97, 103, 109, 164617, 4801, 127, 1854365518528513, 139, 331777, 151, 157, 163, 1652195329, 10093, 181, 9678800287193699463169, 193

* The smallest prime of the form 3*b^n-1 (for even b): [not minimal prime (start with 2 digits) if b<=3]

5, 11, 17, 23, 29, 431, 41, 47, 53, 59, 1451, 71, 2027, 83, 89, 108086391056891903, 101, 107, 113, 4799, 3*42^2523-1, 131, 137, 6911, 149, 8111, 8747, 167, 173, 179, 16913400588503030024793898903900960521239102670648159766677517992069347477035908686646316997626793866297343, 191

* The smallest prime of the form b^n+3 (for b == 2, 4 mod 6): [not minimal prime (start with 2 digits) if either b<=3 or b is prime, but still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

5, 7, 11, 13, 17, 19, 23, 487, 29, 31, 32771, 37, 41, 43, 47, 1799519816997495209117766334283779, 53, 2707, 59, 61, 3847, 67

* The smallest prime of the form b^n-3 (for b == 2, 4 mod 6): [not minimal prime (start with 2 digits) if b<=3]

5, 13, 5, 7, 11, 13, 17, 19, 23, 296196766695421, 29, 31, 54869, 37, 41, 43, 47, 1514785299052682515540398802570879414320893571359760514960122067313271212237031712057484726921232170496646835505906834446399053647478565523037279529736578428914328808517619293356029, 53, 3361, 59, 61

[sweety439]

Originally Posted by sweety439

Base b minimal primes (start with 2 digits) includes:

* The smallest repunit prime base b if exists
* The smallest generalized Fermat prime base b for even b if exists
* The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists
* The smallest Williams prime with 1st kind base b if exists
* The smallest Williams prime with 2nd kind base b if exists
* The smallest Williams prime with 4th kind base b for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest dual Williams prime with 1st kind base b if exists
* The smallest dual Williams prime with 2nd kind base b for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest dual Williams prime with 4th kind base b for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form 2*b^n+1 for bases b>2 if exists
* The smallest prime of the form 2*b^n-1 for bases b>2 if exists
* The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists
* The smallest prime of the form 3*b^n+1 for bases b>3 if exists
* The smallest prime of the form 3*b^n-1 for bases b>3 if exists
* The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists
* The smallest prime of the form 4*b^n+1 for bases b>4 if exists
* The smallest prime of the form 4*b^n-1 for bases b>4 if exists
* The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists
...
* The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Sierpinski conjecture for fixed 1<=k<=b-1) if exists
* The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Riesel conjecture for fixed 1<=k<=b-1) if exists
* The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...) (i.e. the prime for the extended Riesel conjecture for fixed k satisfying these two conditions) if exists
* The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1

The corresponding families:

* repunit prime base b: {1}
* generalized Fermat prime base b for even b: 1{0}1
* generalized half Fermat prime (> (b+1)/2) base b for odd b: {x}y, x = (b-1)/2, y = (b+1)/2
* Williams prime with 1st kind base b: x{y}, x = b-2, y = b-1
* Williams prime with 2nd kind base b: x{0}1, x = b-1
* Williams prime with 4th kind base b: 11{0}1 (not minimal prime if there is smaller prime of the form 1{0}1
* dual Williams prime with 1st kind base b: {x}1, x = b-1
* dual Williams prime with 2nd kind base b: 1{0}x, x = b-1
* dual Williams prime with 4th kind base b: 1{0}11 (not minimal prime if there is smaller prime of the form 1{0}1
* prime of the form 2*b^n+1 for bases b>2: 2{0}1
* prime of the form 2*b^n-1 for bases b>2: 1{x}, x = b-1
* prime of the form b^n+2 for bases b>2 with gcd(b,2)=1: 1{0}2
* prime of the form b^n-2 for bases b>2 with gcd(b,2)=1: {x}y, x = b-1, y = b-2
* prime of the form 3*b^n+1 for bases b>3: 3{0}1
* prime of the form 3*b^n-1 for bases b>3: 2{x}, x = b-1
* prime of the form b^n+3 for bases b>3 with gcd(b,3)=1: 1{0}3
* prime of the form b^n-3 for bases b>3 with gcd(b,3)=1: {x}y, x = b-1, y = b-3
* prime of the form 4*b^n+1 for bases b>4: 4{0}1
* prime of the form 4*b^n-1 for bases b>4: 3{x}, x = b-1
* prime of the form b^n+4 for bases b>4 with gcd(b,4)=1: 1{0}4
* prime of the form b^n-4 for bases b>4 with gcd(b,4)=1: {x}y, x = b-1, y = b-4
* prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Sierpinski conjecture for fixed 1<=k<=b-1): k{0}1
* prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Riesel conjecture for fixed 1<=k<=b-1): x{y}, x = k-1, y = b-1
* prime of the form b^n+k for fixed 1<=k<=b-1: 1{0}k
* prime of the form b^n-k for fixed 1<=k<=b-1: {x}y, x = b-1, y = b-k
* prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1: x{y}, x = (k-1)/gcd(k-1,b-1), y = (b-1)/gcd(k-1,b-1)
* prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1: x = (b-1)/gcd(k-1,b-1), y = (b-k)/gcd(k-1,b-1)

[sweety439]

This puzzle is an extension of the original minimal prime base b puzzle, to include CRUS Sierpinski/Riesel conjectures base b with k-values < b, i.e. the smallest prime of the form k*b^n+1 and k*b^n-1 for all k < b

Also include the dual Sierpinski/Riesel conjectures (of course in the dual case, gcd(k,b) = 1 is needed) base b with k-values < b, i.e. the smallest prime of the form b^n+k and b^n-k for all k < b

[sweety439]

Originally Posted by sweety439

* The smallest prime of the form b^n-2 (for odd b):

7, 3, 5, 7, 14639, 11, 13, 24137567, 17, 19, 480250763996501976790165756943039, 23, 727, 839, 29, 31, 1223, 1367, 37, 2825759, 41, 43, 2207, 47, 45767944570399, 7890479, 53, 1176246293903439667999, 12117359, 59, 61

The b^n-2 case should require n>1, since single-digit primes are not acceptable in this puzzle, thus the smallest primes should be:

7, 23, 47, 79, 14639, 167, 223, 24137567, 359, 439, 480250763996501976790165756943039, 6103515623, 727, 839, 29789, 1087, 1223, 1367, 2313439, 2825759, 1847, 1532278301220703123, 2207, 2399, 45767944570399, 7890479, 3023, 1176246293903439667999, 12117359, 3719, 3967

and the OEIS sequence for the exponent (n) should be A250200, not A255707

Also the b^2-3 case (also should require n>1):

5, 13, 61, 97, 193, 4093, 397, 113379901, 673, 296196766695421, 1021, 1153, 54869, 1597, 1933, 2113, 476837158203124999999999999999999997, 1514785299052682515540398802570879414320893571359760514960122067313271212237031712057484726921232170496646835505906834446399053647478565523037279529736578428914328808517619293356029, 29334891491018187280695810850813, 3361, 916132829, 4093

[sweety439]

Originally Posted by sweety439

All known minimal primes (start with b+1) in bases 2<=b<=16: (data for bases 2, 3, 4, 5, 6, 8, 10, 12 are known to be complete)

Also three unsolved families are known:

Base 11: 5777...777
Base 13: 9555...555
Base 16: DBBB...BBB

DB₃₂₂₃₄ (base 16) is probable prime!!!

Its formula is (206*16^32234-11)/15

This number is like the largest minimal prime (start with 2 digits) in base 16

The families 5{7} (base 11) and 9{5} (base 13) still no (probable) prime found.

The formulas of these two families are (57*11^n-7)/10 and (113*13^n-5)/12, respectively.

[sweety439]

Update newest pdf file.

Minimal elements for the base b representations of the primes which are > b

[Uncwilly]

Originally Posted by sweety439

Update newest pdf file.

Since you have mod rights in this area, you can delete the previous PDF and replace it with the current. Keeping an up to date file and the first post of the thread is a common way of handling things like this. There is no need to post about a new item that will then be added to the pdf. Just update the file. Then maybe 1 time a week give a one line summary for each type of item updated. Less stuff to search through and better organization might make this more useful.