Menu Close

Let-a-b-Z-0-lt-a-lt-b-How-would-you-find-the-maximum-largest-prime-gap-in-a-b-Note-Prime-gaps-are-the-distance-between-consecutive-primes-e-g-7-and-11-has-a-prime-gap-4-p-k-P-p-x-p




Question Number 21292 by FilupS last updated on 19/Sep/17
Let  a,b∈Z  0<a<b     How would you find the maximum/  largest prime gap in (a, b)?    Note:  Prime gaps are the distance between  consecutive primes.  e.g.   7 and 11 has a prime gap 4     p_k ∈P  ∴∀p_x ∀p_(x+1) ∈(a,b):p_(x+1) >p_x   p_(x+1)  and p_x  are consecutive primes  Lets denote δ_x =p_(x+1) −p_x  as prime gap     for (1, 20), the primes are 2,3,5,7,11,13,17  The prime gaps are:  1,2,2,4,2,4  Therefore the largest δ = 4     Is there a more general method?
$$\mathrm{Let}\:\:{a},{b}\in\mathbb{Z} \\ $$$$\mathrm{0}<{a}<{b} \\ $$$$\: \\ $$$$\mathrm{How}\:\mathrm{would}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}/ \\ $$$$\mathrm{largest}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{in}\:\left({a},\:{b}\right)? \\ $$$$ \\ $$$$\mathrm{Note}: \\ $$$$\mathrm{Prime}\:\mathrm{gaps}\:\mathrm{are}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{consecutive}\:\mathrm{primes}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:\mathrm{7}\:\mathrm{and}\:\mathrm{11}\:\mathrm{has}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{4} \\ $$$$\: \\ $$$${p}_{{k}} \in\mathbb{P} \\ $$$$\therefore\forall{p}_{{x}} \forall{p}_{{x}+\mathrm{1}} \in\left({a},{b}\right):{p}_{{x}+\mathrm{1}} >{p}_{{x}} \\ $$$${p}_{{x}+\mathrm{1}} \:\mathrm{and}\:{p}_{{x}} \:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{primes} \\ $$$$\mathrm{Lets}\:\mathrm{denote}\:\delta_{{x}} ={p}_{{x}+\mathrm{1}} −{p}_{{x}} \:\mathrm{as}\:\mathrm{prime}\:\mathrm{gap} \\ $$$$\: \\ $$$$\mathrm{for}\:\left(\mathrm{1},\:\mathrm{20}\right),\:\mathrm{the}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17} \\ $$$$\mathrm{The}\:\mathrm{prime}\:\mathrm{gaps}\:\mathrm{are}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{2},\mathrm{4},\mathrm{2},\mathrm{4} \\ $$$$\mathrm{Therefore}\:\mathrm{the}\:\mathrm{largest}\:\delta\:=\:\mathrm{4} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{more}\:\mathrm{general}\:\mathrm{method}? \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *