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Euler Totient Result

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  • Sebastian Martin
    Can anyone prove the following result?   THEOREM:   Let  p(n-1), p(n), p(n+1) three consecutive prime numbers.   We have:   1/2
    Message 1 of 2 , Jul 8, 2008
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      Can anyone prove the following result?
       
      THEOREM:
       
      Let  p(n-1), p(n), p(n+1) three consecutive prime numbers.
       
      We have:
       
      1/2<  Phi(p(n+1)+pn)/Phi(pn+p(n-1)) <=2
       
      Phi is the Euler Totient function.
       
      Additional Questions :
       
      1)      Can you caracterize the primes that take the integers values 1 and 2?
      2)      There is attain the value 1/2?
       
       
      Sincerely:
       
      Sebastián Martín Ruiz


      ______________________________________________
      Enviado desde Correo Yahoo! La bandeja de entrada más inteligente.

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    • Jens Kruse Andersen
      ... I suggest you get a math program (like PARI/GP used below) and do some quick experimenting before publishing alleged theorems. ? {for(n=2,1000,
      Message 2 of 2 , Jul 8, 2008
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        Sebastian Martin wrote:
        > Can anyone prove the following result?
        > THEOREM:
        > 1/2< Phi(p(n+1)+pn)/Phi(pn+p(n-1)) <=2

        I suggest you get a math program (like PARI/GP used below)
        and do some quick experimenting before publishing alleged theorems.

        ? {for(n=2,1000,
        a=eulerphi(prime(n+1)+prime(n));
        b=eulerphi(prime(n)+prime(n-1));
        r=a/b;
        if(r <= 1/2 | r > 2,
        print(prime(n-1)", "prime(n)", "prime(n+1)": "a"/"b" = "r*1.0)
        ))}
        653, 659, 661: 320/640 = 0.500000000000000000
        1319, 1321, 1327: 1320/640 = 2.06250000000000000
        2297, 2309, 2311: 960/1932 = 0.496894409937888199
        2333, 2339, 2341: 1152/2304 = 0.500000000000000000
        2339, 2341, 2347: 2336/1152 = 2.02777777777777778
        2963, 2969, 2971: 1440/2964 = 0.485829959514170040
        3881, 3889, 3907: 3896/1728 = 2.25462962962962963
        5273, 5279, 5281: 2560/5272 = 0.485584218512898331
        5443, 5449, 5471: 2304/4656 = 0.494845360824742268
        5849, 5851, 5857: 5852/2880 = 2.03194444444444444
        6263, 6269, 6271: 2880/5760 = 0.500000000000000000
        6269, 6271, 6277: 6272/2880 = 2.17777777777777778
        7417, 7433, 7451: 7320/3600 = 2.03333333333333333
        7583, 7589, 7591: 3520/7584 = 0.464135021097046414
        7589, 7591, 7603: 7420/3520 = 2.10795454545454545
        7741, 7753, 7757: 3680/7560 = 0.486772486772486773

        --
        Jens Kruse Andersen
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