# 2 times a prime plus one equals a prime

Discussion in 'General Math' started by Kees Boer, Feb 8, 2006.

1. ### Kees BoerGuest

Hi, I understand that there is a mathemimatical sequence of a certain prime
number that you can multiply it by 2 and add 1 and you'll always get another
prime number and that you can do this indefinitely.

For instance it is not 2, because

2 x 2 = 4 4 + 1 = 5 (prime)
2 x 5 = 10 10 + 1 = 11 (prime)
2 x 11 = 22 22 + 1 = 23 (prime)
2 x 23 = 46 46 + 1 = 47 (prime)
2 x 47 = 94 94 + 1 = 95 ( not a prime)

Sequence for 3 is: 3, 7, 15 (doesn't work)

Sequence for 5 is above minus the first equation. (doesn't work)

Sequence for 7 is: 7, 15 (doesn't work)

Sequence for 9 is: 9, 19, 39 (doesn't work)

I figured out that the first number has to end in the digit 9, because since
I tried 2, 3, 5, 7, and 9 and none of them work. After that, primes only end
in 1, 3, 7, and 9. Well, the 1, when doubled and 1 added will give me as
last digits, 3, then 7, then 5, hence divisible by 5. The 3, will follow the
same pattern, but then 7, and then 5 as last digit. And the 7 will always
have a 5 as the next number. Thus it has to end in 9, which will always end
in a 9 following afterwards.

What is the first prime number in this sequence?

Kees

Kees Boer, Feb 8, 2006

2. ### Don TaylorGuest

That would seem to be a very surprising accomplishment.
If I haven't made a mistake,
For n<= 20,000,000 there are only 9 numbers that keep this up
for more than 6 elements. However, it does seem to give some
support to your conclusion that they must all end in '9'.

{1122659, 2245319, 4490639, 8981279, 17962559, 35925119, 71850239}
{2164229, 4328459, 8656919, 17313839, 34627679, 69255359, 138510719}
{2329469, 4658939, 9317879, 18635759, 37271519, 74543039, 149086079}
{10257809, 20515619, 41031239, 82062479, 164124959, 328249919, 656499839}
{10309889, 20619779, 41239559, 82479119, 164958239, 329916479, 659832959}
{12314699, 24629399, 49258799, 98517599, 197035199, 394070399, 788140799}
{14030309, 28060619, 56121239, 112242479, 224484959, 448969919, 897939839}
{14145539, 28291079, 56582159, 113164319, 226328639, 452657279, 905314559}
{19099919, 38199839, 76399679, 152799359, 305598719, 611197439, 1222394879, 2444789759}

These sequences aren't as uncommon as I might have guessed.
But they are a long way from an unending sequence of primes.

Don Taylor, Feb 8, 2006

3. ### a1jrjGuest

You are correct that modulo 10 a_0 (the first number) has to be 9 as
you have proven above by case by Kees (sorry) examination. You can do
this for other modulo, eg for 12 only 11 works.

But the sequence cant go on indefinitely. the Kth number is a_k = 2^k *
(a_0 + 1) - 1.

Modulo a_0 this is 2^k - 1, and since we require a_0 to be prime, then
by Fermat's little theorem we have
2^(a_0-1) - 1 == 0 mod a_0,

ie the sequence cannot go on longer than a_0 - 1 after which it is
divisible by a_0. In practice the lengths seem to be much shorter but
anyway not infinite.

JJ

a1jrj, Feb 8, 2006
4. ### Don TaylorGuest

Up to 100,000,000 there are only a few more sequences of length >6
and they seem to be becoming less common, as would be expected.

{23103659, 46207319, 92414639, 184829279, 369658559, 739317119, 1478634239}
{24176129, 48352259, 96704519, 193409039, 386818079, 773636159, 1547272319}
{28843649, 57687299, 115374599, 230749199, 461498399, 922996799, 1845993599}
{37088729, 74177459, 148354919, 296709839, 593419679, 1186839359, 2373678719}
{38199839, 76399679, 152799359, 305598719, 611197439, 1222394879, 2444789759}
{42389519, 84779039, 169558079, 339116159, 678232319, 1356464639, 2712929279}
{49160099, 98320199, 196640399, 393280799, 786561599, 1573123199, 3146246399}
{50785439, 101570879, 203141759, 406283519, 812567039, 1625134079, 3250268159}
{52554569, 105109139, 210218279, 420436559, 840873119, 1681746239, 3363492479, 6726984959}
{62800169, 125600339, 251200679, 502401359, 1004802719, 2009605439, 4019210879}
{68718059, 137436119, 274872239, 549744479, 1099488959, 2198977919, 4397955839}
{85864769, 171729539, 343459079, 686918159, 1373836319, 2747672639, 5495345279, 10990690559, 21981381119}
{88174049, 176348099, 352696199, 705392399, 1410784799, 2821569599, 5643139199}
{95831189, 191662379, 383324759, 766649519, 1533299039, 3066598079, 6133196159}

Don Taylor, Feb 8, 2006
5. ### Jasen BettsGuest

there is no such sequence.

Bye.
Jasen

Jasen Betts, Feb 9, 2006