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-   -   Always composite numbers? (https://www.mersenneforum.org/showthread.php?t=25233)

 sweety439 2020-02-14 13:06

Always composite numbers?

Can a dozenal (duodecimal, base 12) number {111...111}21{111...111} (with the same number of 1s in the two brackets) (start with 21, 1211, 112111, 11121111, ...) be prime? I cannot find such prime with <=1000 (decimal 1728) digits, but find neither covering set nor algebra factors. (I have proved that dozenal (duodecimal, base 12) numbers 1{555...555}1 (start with 151, 1551, 15551, 155551, ...) cannot be primes, because of covering sets and algebra factors)

Also, can a dozenal (duodecimal, base 12) 414141...4141411 (start with 411, 41411, 4141411, 414141411, ...) be prime? I want to find the dozenal (duodecimal, base 12) analog of [URL="https://oeis.org/A086766"]A086766[/URL], and I proved that there are no primes for n = 10 (decimal 12) and n = 33 (decimal 39), because of algebra factors, thus the conjecture in [URL="https://oeis.org/A086766"]A086766[/URL] is not true in dozenal (duodecimal, base 12).

Besides, I want to find the dozenal (duodecimal, base 12) analog of many other OEIS sequences, such as [URL="https://oeis.org/A088782"]A088782[/URL], [URL="https://oeis.org/A069568"]A069568[/URL], [URL="https://oeis.org/A200065"]A200065[/URL], [URL="https://oeis.org/A272232"]A272232[/URL], [URL="https://oeis.org/A089776"]A089776[/URL] (n%12 = 1, 5, 7, 11 instead of n%10 = 1, 3, 7, 9), [URL="https://oeis.org/A267720"]A267720[/URL], [URL="https://oeis.org/A244424"]A244424[/URL], [URL="https://oeis.org/A262300"]A262300[/URL], [URL="https://oeis.org/A046035"]A046035[/URL], [URL="https://oeis.org/A047777"]A047777[/URL], [URL="https://oeis.org/A060421"]A060421[/URL], [URL="https://oeis.org/A064118"]A064118[/URL].

For [URL="https://oeis.org/A069568"]A069568[/URL] case, I found the prime (12^1676*298-1)/11 for n = 23 (decimal 27) and proved that there are no primes for n = 34, 89, and 99 (decimal 40, 105, and 117), because of covering sets and algebra factors, but for the [URL="https://oeis.org/A089776"]A089776[/URL] case, I cannot find prime for n = 65 (decimal 77) and n = EE (decimal 143).

 LaurV 2020-02-15 03:55

Are you trolling, or what?

[CODE]
gp > doze(n)=((12^n-1)/11)*(12^(n+2)+1)+25*12^n
%1 = (n)->((12^n-1)/11)*(12^(n+2)+1)+25*12^n
gp > doze(0)
%2 = 25
gp > doze(1)
%3 = 2029
gp > 12^3+13+288
%4 = 2029
gp > doze(2)
%5 = 273181
gp > n=0;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...0...
%6 = 1 [/CODE]HUH??? :shock:

(this fast stop was really unexpected, haha, and it totally took me by surprise.
I should have known that 2029 is prime!
But somewhere in the corner of my mind I was trusting you that there are no primes...

Big mistake...

Anyhow, continuing...

[CODE] gp > isprime(2029)
%7 = 1
gp > n=2;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
%8 = 2
[/CODE] HUH??? :shock: Again?

continuing...
[CODE]
gp > isprime(273181)
%9 = 1
gp > 12^5+12^4+2*12^3+12^2+12^1+1
%10 = 273181
gp > n=3;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
%11 = 3
gp > doze(3)
%12 = 39109981
gp > isprime(%)
%13 = 1
[/CODE]HUH??? :shock: Again?

continuing...
[CODE]
gp > n=4;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...26...
time = 11 ms.
%14 = 27
gp > doze(27)
%15 = 247034206444509265352361785372634788125166592897428447392861
gp > isprime(%)
%16 = 1
gp >
[/CODE]GRRRRRR....

Edit, modify the oneliner to be easier to call (just pressing up arrow, without changing the start value)
[code]
gp > 27
%18 = 27
gp > n=%+1;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...34...
%19 = 35
gp > n=%+1;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...41...
%20 = 42
gp > n=%+1;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...65...
%21 = 66
gp > n=%+1;while(!isprime(doze(n)),printf("...%d...%c",n,13);n++);n
...84...
%22 = 85
gp > doze(85)
%23 = 3782523501323864273640948080633865577531395800266304515211581987031320387343953524514660357926542631809629293255791000553
6420609584690895592018538198985334828642857876896842543151543389
gp > ?digits
digits(x,{b=10}): gives the vector formed by the digits of x in base b (x and b integers).
gp > digits(doze(85),12)
%24 = [
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[/code]Be my guest to count if the form is right... (to make it easyer for you, I aligned and eliminated the spaces using editor's facilities.

Eight primes or so in the first 100... you either were dreaming during testing, or you are trolling.

 axn 2020-02-15 05:18

Simpler form (note: off by 1 from LaurV):
[CODE]doze(n)=(12^(2*n)-1)/11 + 12^n
for(i=1, 2000, n=doze(i); if(ispseudoprime(n), print(i)))
2
3
4
28
36
43
67
86
218
427
1284
[/CODE]

 LaurV 2020-02-15 06:14

yeah, well.. the second form, which I rendered like below (probably a simpler form still exists for this one too) has no primes in sight, but there is no reason why a prime should not be there, the factoring looks like our crus lists, haha, and we would need a sieving program for it, because, with 19 dividing every third, 11 and 17 dividing every 8th, 10th, etc, the chances of finding a prime is very slim, but they should still come in [U]an infinite amount[/U], unless some aurifeuillian factorization unknown to us, or auriferous or platinodiamantine or whatever is called. By construction, these numbers are not divisible by 2, 3 (obviously, they are 1 (mod 12)), and 5, 7 (not so obvious, but see below) and that makes it harder to find a prime. But the prime is there, well hidden..

[CODE]
gp > ?doze41
doze41 = (n)->49*12*(12^(2*n)-1)/143+1
gp > for(i=1,10,print(digits(doze41(i),12)))
[4, 1, 1]
[4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
gp > for(n=1,100,print(n": "factorint(doze41(n))~[1,]))
1: [19, 31]
2: [11, 23, 337]
3: [17, 722237]
4: [13, 19, 1979, 3617]
5: [571, 991, 449929]
6: [177823, 206171347]
7: [19, 277859424790111]
8: [37, 131, 13159, 105173, 113329]
9: [2837, 38587299124605977]
10: [19, 17189, 48268299300277811]
11: [17, 133530286335052494259517]
12: [367, 37560247, 23713556318104789]
13: [11, 19, 23, 35153, 1993859951, 139708400149]
14: [41, 21070110346361, 7846310875960621]
15: [1109, 880130606633990466566760020441]
16: [19, 31, 53, 21023, 78234938983, 2737502719302277]
17: [13, 37, 998861, 279663973627, 150631914918952787]
18: [101, 797, 10891, 1328911, 4006823, 624342125794134911]
19: [17, 19, 5527, 231898913, 2642684909, 383612326598340557]
20: [189767, 2958811, 96442165805129, 1116056700437206537]
21: [8273, 88379, 35125823, 338855864639826302126776175129]
22: [19, 823, 12541, 6390436308030046891653188795484059219533]
23: [3725522325449, 48438583726617962727111389675540810021]
24: [11, 23, 67, 12563393, 505124491, 9544230624930767, 25310427402129151]
25: [19, 83, 643, 1789, 139939, 161312028629, 91378718923313704966951677181]
26: [37, 463, 877, 2387291, 121589407, 123561358204325220849241943959288199]
27: [17, 4564357509721325038561737361681120186114742434887025935677]
28: [19, 1871, 100696837, 3667879961, 851006222623598233419959001107582844037]
29: [36955415772205820717, 1524981620206945422383, 28550311910196094306159]
30: [13, 14718871, 40827524265371788247316943, 29658224926068953911305277722409]
31: [19, 31, 71, 2339, 1469022047, 12220606596293613370899217920874878772972292555153]
32: [242276653, 415099039, 47772475059814551788252488123598816293657760084470783]
33: [50591, 55762433, 121017041968289563417, 2026478573464457245639635066584383583179]
*** factorint: user interrupt after 15,460 ms
*** Break loop: <Return> to continue; 'break' to go back to GP prompt
break>

[/CODE]

 LaurV 2020-02-15 07:55

back to it, here is your (pseudo)prime, which took about 29 minutes to find (I let a "while is not pseudoprime" running, but busy around, didn't check for a while):

[CODE]
gp > #digits(doze41(4383),10)
= 9461
gp > ispseudoprime(doze41(4383))
time = 13,595 ms.
= 1
gp >
[/CODE]

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