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 <x <and <1 $ 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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numerical integration – Problem with NIntegrate in Mathematica 12

I recently installed Mathematica 12 on my laptop and it seems that a particular piece of code that worked in version 10 is generating an error. I added the piece of code. However, the code takes the imported data, so users have no way to prove it. Although, your opinions will help you.

k[m_]: = (7. * 10 ^ 22) /m^0.5;
sigma2[m_]: = (16./81) * Nominate[q^-1*Exp[-q^2/k[m]^ 2]* (q * k[m]^ -1) ^ 4 * PSpectraFunc[q], {q, 10,10 ^ 20}, Method -> {Automatic, "Symbol processing" -> 0}];
betaforma[m_,deltaC_]: = 0.5 * Erfc[deltaC/Sqrt[2sigma2[deltaC/Sqrt[2sigma2[deltaC/Sqrt[2*sigma2[deltaC/Sqrt[2*sigma2[m]]];

The PSpectraFunc function[q] in the NIntegrate it is an interpolation function that uses data that I can not load. However, when I apply the betaform function as

MassRange = Table[i,{i,10^Range[16.,18.,2./100]}];
betaforma[massRange,0.403]

The results show a table containing 0.5 * Erfc[ComplexInfinity]. I had no problems when I ran this in version 10. In version 12, it works when I delete the symbolic processing option, which makes the code inefficient.

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