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@cpsc.ucalgary.ca> To: Althoefer <320045884400-0001@t-online.de> Cc: Richard Guy <rkg@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.