Is there a maximum? $ m en mathbb {N} $ such that for some prime $ p $, $ p_r = p + displaystyle sum_ {n = 1} ^ r 2 ^ n $ is a cousin to $ r = 1,2, …, m $?
There is probably no simple (or known) answer to this question, but I thought it was worth asking.
So far, I have concluded that $ p $ must necessarily have $ 7 $ as its first digit, that is, $ p = 7 , ( text {mod} , 10) $. Yes $ p = x , ( text {mod} , 10) $ for some $ x in {1,3,9 } $, the process is cut in some $ p_k = 5 , ( text {mod} , 10) $.
For example, yes $ p = 1 , ( text {mod} , 10) $, begin {align} p & = 1 , ( text {mod} , 10) \ p_1 & = 3 , ( text {mod} , 10) \ p_2 & = 5 , ( text {mod} , 10) Rightarrow p_2 notin mathbb {P} end {align}
by $ p = 7 , ( text {mod} , 10) $, we have
begin {align} p_ {4k} & = 7 , ( text {mod} , 10) \ p_ {4k + 1} & = 9 , ( text {mod} , 10) \ p_ {4k + 2} & = 3 , ( text {mod} , 10) \ p_ {4k + 3} & = 1 , ( text {mod} , 10) end {align}
for $ k in mathbb {N} cup {0 } $ (Let's suppose for simplicity that $ p_0 = p $).
Consider these two statements:
begin {align} existence p in mathbb {P} &: ; p_r = p + sum_ {n = 1} ^ r 2 ^ n in mathbb {P} ; ; forall r in mathbb {N} tag {1} \ exists m in mathbb {N} &: ; text {for some $ p in mathbb {P} $}, ; p_r = p + sum_ {n = 1} ^ r 2 ^ n in mathbb {P} ; ; forall r in {1,2, …, m } tag {2} \ & ; ; ; ; , text {and} ; forall p in mathbb {P}, ; p_ {m + 1} notin mathbb {P} end {align}
My strategy is simple: try $ (1) Rightarrow (2) $ false or test $ (2) Rightarrow (1) $ false.
Through some computer work, I found that $ m ge $ 9 for $ p = 2397347207 $:
2397347207, 2397347209, 2397347213, 2397347221, 2397347237, 2397347269, 2397347333, 2397347461, 2397347717, 2397348229
If someone can resolve this and / or direct me to useful material, please post it.
