the problematical A126937

Mon Mar 19 11:45:00 CET 2007

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Date: Mon, 19 Mar 2007 10:38:51 -0500
From: xordan <xordan.tom at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: A126933
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 Professor Sloane has written to me:
>>>>>>>>>>>>>>
Dear Tomas Xordan

You submitted this:

%S A126933 2,3,5,11,17
%N A126933 Primes p with property that for every way of writing p =3D a+b,
a^(2^k) + b^(2^k) is a\
 prime for some k > 1.

But do you have a proof that 7 (or 13) is not a member?

If not then the sequence is really  2,3,5, ...

I am going to delete it

(In fact 7 is a member, and probably this is the same as sequence A000040)

Neil Sloane
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Dear Professor Sloane;

I apologize for the mistakes in the post of the sequence 126933.

In fact, I committed two mistakes, the first one when did not include
7 inthe sequence and also as for the exponent (2^n); n> 1 .-

The first (*) mistake I do not Undertand the reason in order that one has
slid, since in my calculations it turns out to be clear that 7 it is in the
sequence :

 7=3D(6+1)=3D(5+2)=3D(4+3)

and :

37=3D(6^2+1^2)

29=3D(5^2+2^2)

337=3D(4^4+3^4) .-

About 13, I find:

13 =3D (12 + 1) ; 12 ^ ?? + 1 ^ ?? =3D  ??

13   =3D (11 + 2); (11 ^ 4 + 2 ^ 4)  =3D  14657

13    =3D(10 + 3); (10 ^ 2 + 3 ^ 2 ) =3D  109

13    =3D(9 + 4 ); (9 ^ 2 + 4 ^ 2)  =3D  97

13    =3D(8 + 5 ); (8 ^ 2 + 5 ^ 2 ) =3D  89

13    =3D(7 + 6 ); (7 ^ 4 + 6 ^ 4)  =3D  3697

In the case of (12+1) I have tried with powers of 2 from 2 until 2048
(12^2048+1^2048) and do not find a prime as result.

But I cannot say that there should not exist any n> 1 such that 12^n + 1 is
prime.

Then I do not have a proof that 13 is not a member either a proof that 13 is
a member=85

I believe that this it is an opened question that can serve in order other
persons investigate and try to demonstrate one  possibility or another.

"=85probably this is the same as sequence A000040"

When you write these words  fills with enthusiasm me very much for the
meaning that would have this for the mathematics.

Probably you could demonstrate this as a theorem...

With all the respect

XORDAN

Ps:(*)For the second mistake  I find explanation since I wanted to describe
first the sequence including the phrase " n it is a power of  2 bigger than
1  (a^n+b^n), n> 1 " ;and later I found clearer to say (a ^ (2^n) +b ^
(2^n)) but I did not correct n> 0 since the initial case is a^1+b^1=3Da ^
(2^0) +b ^ (2^0) .-

Ps: original in spanish, translated by software.

--=20
xordan at hotmail.com
xordan_co at yahoo.com
xordan.tom at gmail.com

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<div class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 12pt; TEXT-ALIGN: left" a=
lign=3D"left"><span style=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial; mso-ansi=
-language: ES">
<div style=3D"PADDING-RIGHT: 10px; PADDING-LEFT: 10px; PADDING-BOTTOM: 10px=
; PADDING-TOP: 10px">Professor Sloane has written to me:</div></span></div>
<div class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 12pt; TEXT-ALIGN: left" a=
lign=3D"left"><span style=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial; mso-ansi=
-language: ES">>>>>>>>>>>>>>></sp=
an>
</div>
<div class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 12pt; TEXT-ALIGN: left" a=
lign=3D"left"><span style=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial; mso-ansi=
-language: ES">Dear Tomas Xordan<br><br>You submitted this:<br><br>%S A1269=
33 2,3,5,11,17
<br>%N A126933 Primes p with property that for every way of writing p =3D a=
+b, a^(2^k) + b^(2^k) is a\<br> prime for some k > 1.<br><br>But do=
 you have a proof that 7 (or 13) is not a member?<br><br>If not then the se=
quence is really  2,3,5, ...
<br><br>I am going to delete it<br><br>(In fact 7 is a member, and probably=
 this is the same as sequence A000040)<br><br>Neil Sloane</span></div>
<div>>>>>>>>>>>>>>>>>>&g=
t;>>>>>>>>>>>>>>>>>>&=
gt;<br clear=3D"all"></div>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">Dear Professor Sloane;</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">I apologize for the mistakes in the post of t=
he sequence 126933.</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">In fact, I committed two mistakes, the first =
one when did not include 7 in the sequence and also as for the exponent (2^=
n); n> 1 .-
</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">The first (*) mistake I do not Undertand the =
reason in order that one has slid, since in my calculations it turns out to=
 be clear that 7 it is in the sequence :
</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span><span lang=3D"ES-CO"><fon=
t face=3D"Times New Roman">7=3D(6+1)=3D(5+2)=3D(4+3) </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">and :</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">37=3D(6^2+1^2)</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">29=3D(5^2+2^2)</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">337=3D(4^4+3^4) .-</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">About 13, I find:</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 =3D (12 + 1) ;<sp=
an style=3D"mso-tab-count: 1">
 </span>12 ^ ?? + 1 ^ ?? =3D<span style=3D"mso-spacerun: yes">  </span=
>?? </span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 <span style=3D"ms=
o-tab-count: 1">
  </span>=3D (11 + 2); (11 ^ 4 + 2 ^ 4)<span style=3D"mso-spacerun: ye=
s">  </span>=3D<span style=3D"mso-spacerun: yes">  </span>14657 <=
/span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 <span style=3D"ms=
o-tab-count: 1">
  </span><span style=3D"mso-spacerun: yes"> </span>=3D(10 + 3); (=
10 ^ 2 + 3 ^ 2 ) =3D<span style=3D"mso-spacerun: yes">  </span>109 <sp=
an style=3D"mso-spacerun: yes"> </span></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 <span style=3D"ms=
o-tab-count: 1">
  </span><span style=3D"mso-spacerun: yes"> </span>=3D(9 + 4 );<s=
pan style=3D"mso-tab-count: 1"> </span>(9 ^ 2 + 4 ^ 2)<span style=3D"mso-sp=
acerun: yes">  </span>=3D<span style=3D"mso-spacerun: yes">  </sp=
an>97 <span style=3D"mso-spacerun: yes">
 </span></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 <span style=3D"ms=
o-tab-count: 1">
  </span><span style=3D"mso-spacerun: yes"> </span>=3D(8 + 5 );<s=
pan style=3D"mso-tab-count: 1"> </span>(8 ^ 2 + 5 ^ 2 ) =3D<span style=3D"m=
so-spacerun: yes">  </span>89 <span style=3D"mso-spacerun: yes"> =
</span></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">13 <span style=3D"ms=
o-tab-count: 1">
  </span><span style=3D"mso-spacerun: yes"> </span>=3D(7 + 6 );<s=
pan style=3D"mso-tab-count: 1"> </span>(7 ^ 4 + 6 ^ 4)<span style=3D"mso-sp=
acerun: yes">  </span>=3D<span style=3D"mso-spacerun: yes">  </sp=
an>3697 <span style=3D"mso-spacerun: yes">
 </span></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES"> </span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">In the case of (12+1=
) I have tried with powers of 2 from 2 until 2048 (12^2048+1^2048) and do n=
ot find a prime as result.
</span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES"> </span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt 2.75pt; TEXT-ALIGN: lef=
t; tab-stops: 21.75pt 65.75pt 175.75pt" align=3D"left"><span style=3D"FONT-=
SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: ES">But I cannot say tha=
t there should not exist any n> 1 such that 12^n + 1 is prime.
</span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO" s=
tyle=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial"> </span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO" s=
tyle=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial">Then I do not have a proof th=
at 13 is not a member either a proof that 13 is a member=85</span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">I believe that this it is an opened question =
that can serve in order other persons investigate and try to demonstra=
te one <span style=3D"mso-spacerun: yes">
 </span>possibility or another.</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span><span lang=3D"ES-CO"><fon=
t face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO" s=
tyle=3D"FONT-SIZE: 9.5pt; FONT-FAMILY: Arial">"=85probably this is the same=
 as sequence A000040"</span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">When you write these words <span style=3D"mso=
-spacerun: yes"> </span>fills with enthusiasm me very much for the mea=
ning that would have this for the mathematics.
</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">Probably you could demonstrate this as a theo=
rem...</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">With all the respect</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman">XORDAN</font></span></p>
<p class=3D"MsoNormal" style=3D"MARGIN: 0cm 0cm 0pt"><span lang=3D"ES-CO"><=
font face=3D"Times New Roman"> </font></span></p>
<div><span lang=3D"ES-CO" style=3D"FONT-SIZE: 12pt; FONT-FAMILY: 'Times=
 New Roman'; mso-ansi-language: ES-CO; mso-fareast-font-family: 'Ti=
mes New Roman'; mso-fareast-language: ES; mso-bidi-language: AR-SA">Ps:=
(*)For the second mistake=20
<span style=3D"mso-spacerun: yes"> </span>I find explanation since I w=
anted to describe first the sequence including the phrase " n it is a =
power of <span style=3D"mso-spacerun: yes"> </span>2 bigger than 1 <sp=
an style=3D"mso-spacerun: yes">
 </span>(a^n+b^n), n> 1 " ;and later I found clearer to say (a=
 ^ (2^n) +b ^ (2^n)) but I did not correct n> 0 since the initial case <=
span lang=3D"ES-CO" style=3D"FONT-SIZE: 12pt; FONT-FAMILY: 'Times New R=
oman'; mso-ansi-language: ES-CO; mso-fareast-font-family: 'Times Ne=
w Roman'; mso-fareast-language: ES; mso-bidi-language: AR-SA">
is a^1+b^1=3Da ^ (2^0) +b ^ (2^0) .-</span></span></div>
<div><span lang=3D"ES-CO" style=3D"FONT-SIZE: 12pt; FONT-FAMILY: 'Times=
 New Roman'; mso-ansi-language: ES-CO; mso-fareast-font-family: 'Ti=
mes New Roman'; mso-fareast-language: ES; mso-bidi-language: AR-SA"></s=
pan>
 </div>
<div><span lang=3D"ES-CO" style=3D"FONT-SIZE: 12pt; FONT-FAMILY: 'Times=
 New Roman'; mso-ansi-language: ES-CO; mso-fareast-font-family: 'Ti=
mes New Roman'; mso-fareast-language: ES; mso-bidi-language: AR-SA"></s=
pan>
 </div>
<div>Ps: original in spanish, translated by software.</div>
<div> </div>
<div> </div>
<div><br>-- <br><a href=3D"mailto:xordan at hotmail.com">xordan at hotmail.com</a=
><br><a href=3D"mailto:xordan_co at yahoo.com">xordan_co at yahoo.com</a><br><a h=
ref=3D"mailto:xordan.tom at gmail.com">xordan.tom at gmail.com</a> </div>

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