Problem 49783. List the good primes

Created by ChrisR

Solve Later

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 37.5px; transform-origin: 407.5px 37.5px; vertical-align: baseline; "><div style="block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 22.5px; text-align: left; transform-origin: 384.5px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 7.66667px; transform-origin: 14px 7.66667px; unicode-bidi: normal; ">The m<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.225px 7.66667px; transform-origin: 27.225px 7.66667px; unicode-bidi: normal; ">th prime <img src="data:image/png;base64,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" alt="p_m" style="width: 18px; height: 20px;" width="18" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.625px 7.66667px; transform-origin: 76.625px 7.66667px; unicode-bidi: normal; "> is classified as “good” if <img src="data:image/png;base64,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" alt="p_m^2 > p_{m-k}p_{m+k}" style="width: 92.5px; height: 22px;" width="92.5" height="22"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 7.66667px; transform-origin: 12.05px 7.66667px; unicode-bidi: normal; "> for <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAAkCAYAAAAgqxBxAAADeklEQVR4nO2bfZGzMBDGHw84wAAGUICCOsBBHdRCNSChHs4CGs5C3z/CM122oVNIQpJ79zeTubnmSr6e7G42HGAYhmEYhmEYhmEYhvE/0gJocncikA7AAOAGYMzcF+N7upgPawHcATxjP/hEBgAPAL9w43gunxll0+O1bsE0eAmZJbegB4R5CTme3N6mQ/75LJUOTshcq2BBDwCuy88f5Bf0BcAcoQ+cpEeMTh2EVucJC3t8jHhpjx41ioUmV+QTtBTyE8AEFwIdhc+5hndtN1LITzhD0WfoR01wvqoXtE/IoYvfi+edKSSfkC8ntl8z1Qs6hZCJHMcZmJDDqVLQzdJGKiETngVSx88XrM8dKYXcY324bZbPfOv0qa5UqhI0hSxTaSmEzLbYBg9irWr/B2FZFO1dYguZ/Z1EO3JzynXSackR6wNWLbF7FYI+U8hkgH8MzdKPkENiaiFr5Hqw33e4xZfjvIm6eemTriud4gUtLQUtzBnW4ob3iWngxHd0TD3WQqZoUjNivR53OIMArD3RBW7cvrpa0oXFC1reOkprEZKK+wYKj4vbIjwF6PM0D6SPUTl/9CyTqJOZnBFuw5ItL/UJxt0hJca9QbGCJj4xpBJ2i/Ui07LGvCnUYUdKYcv4eVZ1XKsZbm7lGG+i7lvkBjlaprenfk81giZnCFvGjnLBU1x9pxa23Jy+w51ME+rwh5mX+472eAUdUkLOJ9UJmmwJO0Z7E9YiOCOO9Ak7xllBxs/6YCdj5J8PdTXlwqsVNPHlpEOtHEX8wHph+XtKdE56RthbfnJzai8mY2TdxvDheyVTvaAlMdx3h3fLJA+lZ+Vj9a3hEWHLzeiLS32ZHF2nLXfp/ClBEy3sPX2QLpqWSYp8Un+bWuBa2HvCHmllfWED58gXI9NLyNx0De+D/0lBEwp7Tx84IdoySVENyzP3nP5DobD3CFp6Fo3cpD6hyroR+w6GOYku6BZr65jjtUvNnhhw6wDVYn0V/Is8m3XPWHQuXSI9kT4XyNTbvPH9EtEpwyDv2cKJwJeGuaOOmyaZcvKJtYNb3GmjviRavMbiW1iule9Ku4Eb4wN1ZDh6vK7ydbmhsFCpQ/jNU+5/nSItwsdSU8bB8CBj16OlFGuq3247UmrwdMYHtsKYPaUUq3ZB+FiKcp+GYRiGYRiGYRiGYRhGCfwDVg8cDxGTmBkAAAAASUVORK5CYII=" alt="1 <= k <= m-1" style="width: 90px; height: 18px;" width="90" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.692px 7.66667px; transform-origin: 130.692px 7.66667px; unicode-bidi: normal; ">. For example, 5 is a good prime because <img src="data:image/png;base64,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" alt="5^2 = 25" style="width: 49px; height: 19px;" width="49" height="19"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.6167px 7.66667px; transform-origin: 48.6167px 7.66667px; unicode-bidi: normal; "> is greater than <img src="data:image/png;base64,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" alt="2x11 = 22" style="width: 70px; height: 18px;" width="70" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="3x7 = 21" style="width: 62.5px; height: 18px;" width="62.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.45px 7.66667px; transform-origin: 68.45px 7.66667px; unicode-bidi: normal; ">, but 7 is not because <img src="data:image/png;base64,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" alt="7^2 < 5x11" style="width: 67.5px; height: 19px;" width="67.5" height="19"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; ">. </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.5917px 7.66667px; transform-origin: 93.5917px 7.66667px; unicode-bidi: normal; ">Write a function to list the first n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3417px 7.66667px; transform-origin: 44.3417px 7.66667px; unicode-bidi: normal; "> good primes. </div></div></div>

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Jean-Marie Sainthillier on 23 Nov 2021

In the description, k>=1

ChrisR on 20 Dec 2021

Thanks--fixed.

Christian Schröder on 25 Mar 2023

Why is 2 not a good prime? With m = 1, it trivially holds that p_m^2 > p_{m-k} p_{m+k} for all 1 \le k \le m-1 since this is in fact an empty statement then.

ChrisR on 28 Mar 2023

It's not included in the lists I've seen probably because there is no zeroth prime.

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Problem 49783. List the good primes

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Problem 49783. List the good primes

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