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

• To: mathgroup at smc.vnet.net
• Subject: [mg107046] 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: Mon, 1 Feb 2010 06:13:40 -0500 (EST)

```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 two
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, there
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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