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AP with tau > 10^(10^100)

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  • djbroadhurst
    Primitive AP with same tau 10^(10^100) Phil asked for some numbers and equations, to dilute the metaphysics. Here goes. Construction: Find a pair of titanic
    Message 1 of 1 , Feb 2, 2002
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      Primitive AP with same tau > 10^(10^100)

      Phil asked for some numbers and equations,
      to dilute the metaphysics. Here goes.

      Construction:

      Find a pair of titanic primes (q,r) with q < r and
      sqrt((q^2+r^2)/2) = prod(k=1,315,p[k])
      with p[1] = 5 and primes p[k+1] > p[k].
      Let a[1] = 1 and a[k+1] = a[k]*(a[k]+3)
      m = prod(k=1,315,p[k]^a[316-k])/5
      n = m*q^2
      d = m*(r^2-q^2)/2.
      Let tau(n) denote the number of divisors of n. Then

      tau(n) = tau(n+d) = tau(n+2*d) = a[316] - a[315]

      is a primitive equal-tau arithmetic progression
      with tau = O(c^(2^316)) and n = O(5^(c^(2^315))) where
      c = 1.52652645702134522042544787533286961241905496697...
      Note that tau > 10^(10^100) and n > 10^(10^(10^100)).

      {q=
      96832739653606690002650846957211649965819969564016\
      14429489977777850251873224785387331673907025941001\
      46944666266594551422539786143013670426682693644404\
      19238863257105123347976848172235752041386553093372\
      48535877327910237923784357367080678831157021576325\
      72506457686266198317983845220610991916982385743850\
      60207883746064540588248955268899219769671466643056\
      91679903902064557030090343575322151405760466593069\
      41298828766601021680325601122800469966705299982330\
      55184801612892403828125477722988592786794914288113\
      68649831145277363149052302335240518810293845555580\
      11294844559691006494430396704616024842910321671645\
      32298515898030651179590665799056896119255025023578\
      08307167500572209470735044710301111092252340279504\
      34042525748754592274196130309678764053674304915526\
      29168253065667461189470581137995214054456438205296\
      33904063046321792327536369393219748759424117130312\
      80593689549732362111072244258815319601894124830241\
      23688142711387217893062889667648509933950605485911\
      39833131564742458956575017081404255669669037631821\
      7;
      r=
      58668701629406543657566897844304390566476704731357\
      70070742555390600649317794015252872435379503354750\
      72377442725450476816428518626608146062874302469997\
      72943256593950768935973698480777933026059483393921\
      23205957852709231899380163801085793957364434675415\
      46980100693770045408647880767702846748492842643871\
      95091185828984252296393989236432506007214071134991\
      23943633398632795052802245076990274344583044537012\
      99250718703050953568412324707225690516184809753299\
      57161655400128502233484119575885304193231855104462\
      43229053350537740313521030780890044203874015850969\
      30576691037086133213376918503591949328962436011459\
      60176014762828334125585163528170544961857145080243\
      84483577988357000123548415829300771590068225388060\
      21931316836264587796726652945443534712942473478111\
      62913308799294763497336109551964682376315185785654\
      99838953833911714910302661983598281335852954622481\
      96662152737546402073319285715129044831232993619676\
      09037987589020843515996268909412597942163523672769\
      66811170061195635570758033335593420317129766814427\
      69;
      if(issquare((q^2+r^2)/2,&f)&&!issquare(f),
      print(matsize(factor(f))[1]))}
      315
      Good bye!
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