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## number theory: is there a maximum of \$ m in mathbb {N} \$ such that for some cousins ​​\$ p \$, \$ p_r = p + displaystyle sum_ {n = 1} ^ r 2 ^ n \$ is prime for \$ r = 1,2, …, m \$?

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.

## inequalities – Show that this expression is greater than 1/2

Leave $$0 be given Try out
$$4x ^ {2} + 4y2-4xy-4y + 1 + frac {4} { pi ^ 2} Big[ sin^{2}(pi x)+ sin^{2}(pi y) + sin^{2}[pi(y-x)] Big] geq frac {1} {2}$$

I have been working on this problem for a while and I have hit a wall. I have traced this and it seems to be true. I also tried to prove that it is greater than some other fraction like $$frac {2} {5}$$ or something. So far, I have tried to make partial derivatives with respect to x, but unfortunately, it is very difficult to find the roots, since it involves the equation that involves $$sin (2 pi x)$$, $$sin[2 pi (x-y)]$$ Y $$x$$. I think the expansion of the Taylor series would look very ugly. I would really appreciate it if anyone had any suggestions on what to try or some branch of mathematics that I can read, since I'm not sure how to handle this type of expression, except the elementary tool that I have.

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