Conjecture:at least one prime p between (n-1)n/2 and n(n+1)/2
in [Mathematica]
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From: a boy on 1 Feb 2010 06:13 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[58]:= 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
From: a boy on 3 Feb 2010 06:13 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....(a)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[58]:= 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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