{"id":206,"date":"2011-04-30T05:10:46","date_gmt":"2011-04-30T09:10:46","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=206"},"modified":"2011-04-30T05:10:46","modified_gmt":"2011-04-30T09:10:46","slug":"any-prime-number-is-the-average-of-at-least-one-pairs-of-inequal-prime-numbers","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=206","title":{"rendered":"Any prime number is the average of at least one pairs of inequal prime numbers"},"content":{"rendered":"

According to Goldbach’s conjecture, any 2*p = p + p fulfills it. This can be a little stronger, to make 2*p=p1+p2, p1 <> p2, it still stands.<\/p>\n

For example:<\/p>\n

“Prime[5]:11+\/-6*1=5+17”
\n“Prime[6]:13+\/-6*1=7+19”
\n“Prime[7]:17+\/-6*1=11+23”
\n“Prime[8]:19+\/-6*2=7+31”
\n“Prime[9]:23+\/-6*1=17+29”
\n“Prime[10]:29+\/-6*2=17+41”
\n“Prime[11]:31+\/-6*2=19+43”
\n“Prime[12]:37+\/-6*1=31+43”
\n“Prime[13]:41+\/-6*2=29+53”
\n“Prime[14]:43+\/-6*4=19+67”<\/p>\n

The full list up to:<\/p>\n

“Prime[1306583450]:30158696947+\/-6*185=30158695837+30158698057”<\/p>\n

is downloadable as GoldbachAverage_typed_ranks.primes.tgz<\/a> (caution, large file).<\/p>\n

It is straight forward that only when k=6i, it is possible both p+k and p-k are both prime.<\/p>\n

Assuming for prime p, both p+6i and p-6i are prime, the first occurance of i forms the integer sequence A139602<\/a>, named:<\/p>\n

“a(n) is the smallest prime p that makes the pair p+\/-6n both primes while no other pair of p+\/-6k, 0<k<n primes “.<\/p>\n

Defining\u00a0a(n) = A139602(m) such that for any k>m A139602(k) > A139602(m), the first 65 items are listed below:
\n“Prime[5]:11+\/-6*1=5+17”
\n“Prime[8]:19+\/-6*2=7+31”
\n“Prime[14]:43+\/-6*4=19+67”
\n“Prime[25]:97+\/-6*5=67+127”
\n“Prime[38]:163+\/-6*6=127+199”
\n“Prime[43]:191+\/-6*7=149+233”
\n“Prime[48]:223+\/-6*10=163+283”
\n“Prime[88]:457+\/-6*14=373+541”
\n“Prime[151]:877+\/-6*15=787+967”
\n“Prime[176]:1049+\/-6*17=947+1151”
\n“Prime[214]:1307+\/-6*20=1187+1427”
\n“Prime[300]:1987+\/-6*21=1861+2113”
\n“Prime[308]:2029+\/-6*25=1879+2179”
\n“Prime[320]:2129+\/-6*30=1949+2309”
\n“Prime[577]:4217+\/-6*34=4013+4421”
\n“Prime[853]:6599+\/-6*45=6329+6869”
\n“Prime[1228]:9967+\/-6*51=9661+10273”
\n“Prime[1271]:10357+\/-6*79=9883+10831”
\n“Prime[2090]:18233+\/-6*81=17747+18719”
\n“Prime[6615]:66343+\/-6*89=65809+66877”
\n“Prime[7356]:74573+\/-6*111=73907+75239”
\n“Prime[9243]:95911+\/-6*113=95233+96589”
\n“Prime[9568]:99719+\/-6*132=98927+100511”
\n“Prime[16880]:186551+\/-6*133=185753+187349”
\n“Prime[17686]:196337+\/-6*135=195527+197147”
\n“Prime[18911]:211219+\/-6*157=210277+212161”
\n“Prime[23026]:262469+\/-6*160=261509+263429”
\n“Prime[24229]:277301+\/-6*163=276323+278279”
\n“Prime[35125]:416573+\/-6*175=415523+417623”
\n“Prime[49360]:603487+\/-6*211=602221+604753”
\n“Prime[78101]:994549+\/-6*222=993217+995881”
\n“Prime[107328]:1403137+\/-6*271=1401511+1404763”
\n“Prime[290914]:4117441+\/-6*273=4115803+4119079”
\n“Prime[335833]:4805761+\/-6*290=4804021+4807501”
\n“Prime[341710]:4895789+\/-6*307=4893947+4897631”
\n“Prime[401477]:5823067+\/-6*309=5821213+5824921”
\n“Prime[402723]:5842813+\/-6*341=5840767+5844859”
\n“Prime[521180]:7704409+\/-6*385=7702099+7706719”
\n“Prime[965375]:14911571+\/-6*390=14909231+14913911”
\n“Prime[1041561]:16174121+\/-6*427=16171559+16176683”
\n“Prime[1403631]:22245077+\/-6*460=22242317+22247837”
\n“Prime[2706070]:44786009+\/-6*472=44783177+44788841”
\n“Prime[3165153]:52912507+\/-6*569=52909093+52915921”
\n“Prime[9066474]:161738579+\/-6*627=161734817+161742341”
\n“Prime[11872208]:215189881+\/-6*632=215186089+215193673”
\n“Prime[13761571]:251589509+\/-6*772=251584877+251594141”
\n“Prime[37548968]:726419297+\/-6*791=726414551+726424043”
\n“Prime[45509717]:889697437+\/-6*805=889692607+889702267”
\n“Prime[70323838]:1407132329+\/-6*833=1407127331+1407137327”
\n“Prime[73701270]:1478355583+\/-6*855=1478350453+1478360713”
\n“Prime[91179428]:1849422983+\/-6*864=1849417799+1849428167”
\n“Prime[105481347]:2155728167+\/-6*899=2155722773+2155733561”
\n“Prime[114690246]:2354047967+\/-6*980=2354042087+2354053847”
\n“Prime[126391595]:2607147923+\/-6*986=2607142007+2607153839”
\n“Prime[149035786]:3100157353+\/-6*994=3100151389+3100163317”
\n“Prime[179785305]:3775276711+\/-6*1000=3775270711+3775282711”
\n“Prime[197058908]:4157064299+\/-6*1098=4157057711+4157070887”
\n“Prime[227165893]:4826177711+\/-6*1145=4826170841+4826184581”
\n“Prime[416967557]:9124822669+\/-6*1150=9124815769+9124829569”
\n“Prime[426241552]:9337604803+\/-6*1166=9337597807+9337611799”
\n“Prime[460241741]:10119526379+\/-6*1213=10119519101+10119533657”
\n“Prime[544180839]:12060924103+\/-6*1219=12060916789+12060931417”
\n“Prime[699679610]:15691824767+\/-6*1279=15691817093+15691832441”
\n“Prime[1083294128]:24791915689+\/-6*1292=24791907937+24791923441”
\n“Prime[1295780294]:29898040813+\/-6*1385=29898032503+29898049123”<\/p>\n","protected":false},"excerpt":{"rendered":"

According to Goldbach’s conjecture, any 2*p = p + p fulfills it. This can be a little stronger, to make 2*p=p1+p2, p1 <> p2, it still stands. For example: “Prime[5]:11+\/-6*1=5+17” “Prime[6]:13+\/-6*1=7+19” “Prime[7]:17+\/-6*1=11+23” “Prime[8]:19+\/-6*2=7+31” “Prime[9]:23+\/-6*1=17+29” “Prime[10]:29+\/-6*2=17+41” “Prime[11]:31+\/-6*2=19+43” “Prime[12]:37+\/-6*1=31+43” “Prime[13]:41+\/-6*2=29+53” “Prime[14]:43+\/-6*4=19+67” The full list up to: “Prime[1306583450]:30158696947+\/-6*185=30158695837+30158698057” is downloadable as GoldbachAverage_typed_ranks.primes.tgz (caution, large file). It is straight […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-206","post","type-post","status-publish","format-standard","hentry","category-uncategorized","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=206"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/206\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}