       Solve stuck at 243

• To: mathgroup at smc.vnet.net
• Subject: [mg124206] Solve stuck at 243
• From: Ralph Dratman <ralph.dratman at gmail.com>
• Date: Fri, 13 Jan 2012 04:53:15 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Project Euclid asks, "What is the smallest odd composite
that cannot be written as the sum of a prime and twice a
square?"

I tried the following equation, not really expecting it to
work:

oddComposite == Prime[m] + 2 k^2

Surprisingly, the above actually does work for all the odd
composite numbers through 237.

solveInstance[oddComposite_] := Solve[{oddComposite ==
Prime[m] + 2*k^2, k > 0, m > 0}, {k, m}, Integers];
For[i = 9, i < 300, i = i + 2,
If[Not[PrimeQ[i]], Print[i,": ", solveInstance[i]]]]

9: {{k->1,m->4}}
15: {{k->1,m->6},{k->2,m->4}}
21: {{k->1,m->8},{k->2,m->6},{k->3,m->2}}
25: {{k->1,m->9},{k->2,m->7},{k->3,m->4}}
27: {{k->2,m->8}}
33: {{k->1,m->11}}
35: {{k->3,m->7},{k->4,m->2}}
39: {{k->1,m->12},{k->2,m->11},{k->4,m->4}}
45: {{k->1,m->14},{k->2,m->12},{k->4,m->6}}
49: {{k->1,m->15},{k->2,m->13},{k->3,m->11},{k->4,m->7}}
51: {{k->2,m->14},{k->4,m->8}}

- - - - - - snip - - - - - -

217: {{k->3,m->46},{k->5,m->39},{k->8,m->24},{k->10,m->7}}
219: {{k->2,m->47},{k->10,m->8}}
221: {{k->6,m->35},{k->9,m->17}}
225: {{k->1,m->48},{k->4,m->44},{k->7,m->31},{k->8,m->25}}
231:
{{k->1,m->50},{k->2,m->48},{k->4,m->46},{k->5,m->42},{k->8,m
->27},{k->10,m->11}}
235:
{{k->1,m->51},{k->2,m->49},{k->6,m->38},{k->7,m->33},{k->8,m
->28},{k->9,m->21}}
237: {{k->2,m->50},{k->7,m->34},{k->8,m->29},{k->10,m->12}}

- - - - - - but then, at 243, something changes - - - - -

243: {{k->1,m->53},{k->4,m->47},{k->5,m->44},{k->10,m->14}}
Solve::nsmet: This system cannot be solved with the methods
available to Solve. >>

245: Solve[{245==2 k^2+Prime[m],k>0,m>0},{k,m},Integers]
Solve::nsmet: This system cannot be solved with the methods
available to Solve. >>

247: Solve[{247==2 k^2+Prime[m],k>0,m>0},{k,m},Integers]
Solve::nsmet: This system cannot be solved with the methods
available to Solve. >>

... and so on. Strange.

Does anyone know why such a threshold might appear?

Thank you.

Ralph Dratman

```

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