       Re: Conjecture:at least one prime p between (n-1)n/2 and n(n+1)/2

• To: mathgroup at smc.vnet.net
• Subject: [mg107137] Re: Conjecture:at least one prime p between (n-1)n/2 and n(n+1)/2
• From: a boy <a.dozy.boy at gmail.com>
• Date: Wed, 3 Feb 2010 06:13:50 -0500 (EST)
• References: <hk6d0n\$m4h\$1@smc.vnet.net>

```Clear[n];
FindInstance[ NextPrime[n (n - 1)/2] > n (n + 1)/2 &&   0 < n <
2^2^20, {n}, Integers]

FindInstance::nsmet: The methods available to FindInstance are
insufficient to find the requested instances or prove they do not
exist.

On Feb 1, 7:13 pm, a boy <a.dozy.... at gmail.com> wrote:
> I have a Conjecture: For every n > 1 there is always at least one prime p
> such that (n-1)n/2 < p < n(n+1)/2
>
> n = 2; b = 1;
> While[NextPrime[b] < (b = n (n + 1)/2), n++]
> n
>
> \$Aborted
>
> 14394105
>
> In this diagram,there is at least one small point(prime) between every tw=
o
> medium points(triangle number ).
>
> In:= start = 1;
> n = 400;
> pl = Table[{Prime[i], n}, {i, start, n}];
> tl = Table[{i (i + 1)/2, n}, {i, start, n}];
> ListLinePlot[Prime[Range[start, n]],
>  Epilog -> {PointSize[Medium], Point[tl], PointSize[Small], Point[pl]}]
>
> The still unsolved Legendre's conjecture asks whether for every n > 1, th=
ere
> is a prime p, such that n^2 < p < (n + 1)^2. Comparing,
>
> (n+1)^2-n^2=2n+1, while n(n+1)/2-(n-1)n/2=n

```

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