## Re: Eleven successive primes not full period - is this a record?

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• ... [25, 461413, 461803] [29, 6754837, 6755267] [30, 27214597, 27215039] David
Message 1 of 12 , Aug 8, 2009
Jack Brennen <jfb@...> wrote:

> A few minutes with PARI/GP yields 25 consecutive such primes
> from 461413 to 461803.

[25, 461413, 461803]
[29, 6754837, 6755267]
[30, 27214597, 27215039]

David
• ... This is trivial as others have shown. The next case of eleven is as small as 4481 to 4561. Consecutive primes which *are* full period primes appear more
Message 2 of 12 , Aug 8, 2009
julienbenney wrote:
> 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121
>
> These are eleven consecutive primes not one of which is a full period
> prime.
>
> The question I want to ask is whether there is any other sequence of
> consecutive non-full-period primes equal to or greater than this sequence.

This is trivial as others have shown.
The next case of eleven is as small as 4481 to 4561.
Consecutive primes which *are* full period primes appear more interesting.
In http://www.primepuzzles.net/puzzles/puzz_095.htm I found 23 from
239651440411 to 239651440949.

--
Jens Kruse Andersen
• ... Those are much easier to handle, since one may use a kronecker filter, as Chris Nash observed. I prefer Julien s twist: 10 is not a primitive root of unity
Message 3 of 12 , Aug 8, 2009
"Jens Kruse Andersen" <jens.k.a@...> wrote:

> Consecutive primes which *are* full period primes
> appear more interesting.

Those are much easier to handle, since one may
use a kronecker filter, as Chris Nash observed.

I prefer Julien's twist:

10 is not a primitive root of unity modulo any of
the 33 consecutive primes from 923125117 to 923125939

n=33;p=923125117;v=vector(n);
for(k=1,33,v[k]=(p-1)/znorder(Mod(10,p));p=nextprime(p+1));
print(v);

[6, 2, 2, 4, 2, 30, 6, 4, 7, 3, 6, 2, 7, 2, 8, 3,
2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 12, 3, 2, 4, 4, 7, 7]

Any advance on 33 consecutive primes of this kind?

David
• ... [35, 1292613521, 1292614321] [36, 2757553987, 2757554983] -- Jens Kruse Andersen
Message 4 of 12 , Aug 8, 2009
> Any advance on 33 consecutive primes of this kind?

[35, 1292613521, 1292614321]
[36, 2757553987, 2757554983]

--
Jens Kruse Andersen
• ... 10 is not a primitive root of unity modulo any of the 35 consecutive primes beginning with 1292613521 n=35;p=1292613521;v=vector(n);
Message 5 of 12 , Aug 8, 2009

> Any advance on 33 consecutive primes of this kind?

10 is not a primitive root of unity modulo any of
the 35 consecutive primes beginning with 1292613521

n=35;p=1292613521;v=vector(n);
for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p=nextprime(p+1));
print(v);

[10, 2, 2, 2, 2, 3, 2, 7, 2, 2, 4, 3, 4, 2, 3, 8, 14, 4,
2, 6, 2, 6, 2, 2, 3, 12, 2, 2, 3, 21, 2, 2, 3, 3, 10]

David
• ... Ah, Jens beat my case with 35, which was to be expected. David
Message 6 of 12 , Aug 8, 2009
"Jens Kruse Andersen" <jens.k.a@...> wrote:

> > Any advance on 33 consecutive primes of this kind?
>
> [35, 1292613521, 1292614321]
> [36, 2757553987, 2757554983]

Ah, Jens beat my case with 35, which was to be expected.

David
• ... [37, 7088772637, 7088772653] David
Message 7 of 12 , Aug 8, 2009
"Jens Kruse Andersen" <jens.k.a@> wrote:

> [36, 2757553987, 2757554983]

[37, 7088772637, 7088772653]

David
• ... Whoops! The last of these 37 primes is 7088773519 n=37;p1=7088772637;v=vector(n);p=p1; for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
Message 8 of 12 , Aug 8, 2009

> [37, 7088772637, 7088772653]

Whoops! The last of these 37 primes is 7088773519

n=37;p1=7088772637;v=vector(n);p=p1;
for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
print([n,p1,p2,v]);

[37, 7088772637, 7088773519,
[2, 52, 2, 2, 2, 10, 4, 6, 2, 4, 2, 2, 2, 6, 2, 2, 8, 4,
3, 2, 2, 3, 3, 60, 3, 2, 4, 35, 3, 4, 4, 3, 2, 25, 2, 2, 2]]

With apologies for the typo,

David
• ... 10 is not a primitive root of unity modulo any of the 40 consecutive primes beginning with 22588287443 n=40;p1=22588287443; v=vector(n);p=p1;
Message 9 of 12 , Aug 8, 2009

> 37, 7088772637, 7088773519

10 is not a primitive root of unity modulo any of
the 40 consecutive primes beginning with 22588287443

n=40;p1=22588287443;
v=vector(n);p=p1;
for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
print([n,p1,p2,v]);

[40, 22588287443, 22588288409,
[46, 15, 5, 2, 2, 5, 2, 2, 2, 2, 7, 2, 5, 9, 7, 30, 2, 4, 3, 3,
3, 4, 3, 3, 2, 6, 2, 9, 2, 16, 2, 2, 2, 2, 2, 6, 48, 5, 2, 2]]

David
• ... 10 is not a primitive root of unity modulo any of the 45 consecutive primes beginning with 24124140487 n=45;p1=24124140487; v=vector(n);p=p1;
Message 10 of 12 , Aug 8, 2009

> 10 is not a primitive root of unity modulo any of
> the 40 consecutive primes beginning with 22588287443

10 is not a primitive root of unity modulo any of
the 45 consecutive primes beginning with 24124140487

n=45;p1=24124140487;
v=vector(n);p=p1;
for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
print([n,p1,p2,v]);

[45, 24124140487, 24124141903,
[3, 2, 6, 2, 2, 90, 6, 3, 7, 4, 8, 3, 12, 4, 4,
2, 2, 9, 20, 6, 7, 2, 4, 2, 36, 42, 2, 2, 2, 2,
2, 2, 117, 10, 3, 15, 3, 4, 2, 2, 10, 2, 2, 2, 7]]

David
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