Sums of Primes and Squares (fwd)

[Richard Guy]

Here's an answer I've just made.
Can any munster or sequester comment
on the finiteness or otherwise of
A064233 ?   Best to all,   R.

---------- Forwarded message ----------
Date: Wed, 30 Mar 2005 13:07:06 -0700 (MST)
From: Richard Guy <rkg at cpsc.ucalgary.ca>
To: Althoefer <320045884400-0001 at t-online.de>
Cc: Richard Guy <rkg at cpsc.ucalgary.ca>
Subject: Re: Sums of Primes and Squares

I'm not sure if I can be of much help.

A19 of UPINT doesn't quite cover it.

But let me tackle (ii):-

If  m^2 = n^2 + p, then necessarily
n = m-1  and  p = 2m-1  so all
numbers  [(p+1)/2]^2 are expressible
as  [(p-1)/2]^2 + p.  On the other hand,
if  m  is not of the form  (p+1)/2
then  m^2  is not equal to  r^2 + p
for any  r, because  p = m^2 - r^2 =
(m-r)(m+r) requires  m-r = 1  and
then  p = m+r = 2m-1  which we've
assumed to be false.  So the squares
of 5, 8, 11, 13, 14, 17, 18, 20, 23,
25, 26, 28, 29, ... are not the sum
of a square and a prime.  I.e.,
almost all squares aren't.

As for (i), sequence A064233 doesn't
give any references, but I'd guess
that the sequence is as likely to be
infinite as finite -- good question
which may find its way into
future editions of UPINT.

The answer for (iii) is presumably
the same as that for (i).  Let me
know if you learn more.    R.

ID Number: A064233
URL: 
http://www.research.att.com/projects/OEIS?Anum=A064233
Sequence: 
1,2,5,10,13,25,31,34,37,58,61,64,85,91,121,127,130,169,196,
            214,226,289,324,370,379,400,439,526,529,571,625,676,706,730,
            771,784,829,841,991,1024,1089,1225,1255,1351,1414,1444,1521,
            1549,1681,1849,1906,1936,2116
Name:      Numbers which are not the sum of a prime 
number and a nonzero square.
Example:   5 = 1+4 or 2+3; a prime and a square do not 
appear together in either sum
Math'ca:   Complement[ Table[ n, {n, 1, 10000} ], Union[ 
Flatten[ Table[ Prime[ i ]
               + j^2, {i, 1, 1230}, {j, 1, 100} ] ] ] ]
See also:  Adjacent sequences: A064230 A064231 A064232 
this_sequence A064234
               A064235 A064236
            Sequence in context: A003654 A047617 A018571 
this_sequence A051952
               A103188 A064392
Keywords:  easy,nonn,nice
Offset:    0
Author(s): Axel Harvey (axe(AT)cam.org), Sep 22 2001
Extension: More terms from Vladeta Jovovic 
(vladeta(AT)Eunet.yu), Robert G.
               Wilson v (rgwv(AT)rgwv.com) and Felice 
Russo
               (felice.russo(AT)katamail.com), Sep 23 2001

On Wed, 30 Mar 2005, Althoefer wrote:

> Dear Prof. Guy,
> 
> can you please tell me, what is known about the 
> following questions:
> 
> (i) Are there infinitely many natural numbers which 
> are NOT the sum of a prime
> and a square? (For instance, 130 is such a number; 
> 129=113+16 is not.)
> 
> (ii) Are there infinitely many squares of natural 
> numbers which are NOT the sum
> of a prime and a square? (For instance, 625 is such a 
> number.)
> 
> (iii) Are there infinitely many natural numbers, which 
> are both not squares
> numbers and not the sum of a prime and a square?
> 
> My conjecture is that all three answers should be NO.
> 
> Thanks in advance for your reply.
> Sincerely yours, Ingo Althofer.