5.- Conjecture or theorem ?
in [Math]
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From: Ludovicus on 27 Jul 2010 12:21 "The set of least divisors of x^2 + x + A is the set of prime numbers, (A = an odd integer.) for each A testing x = 0,1,2,3...(Non negative integers)." Example: 79 is the least divisor of x^2 + x + 26149427 . (Its first appearance as minimum divisor) Ludovicus
From: Gerry Myerson on 27 Jul 2010 19:39 In article <ded925ae-4677-4707-8dd3-d4c4a3d7b212(a)x4g2000vbe.googlegroups.com>, Ludovicus <luiroto(a)yahoo.com> wrote: > "The set of least divisors of x^2 + x + A is the set of prime > numbers, (A = an odd integer.) > for each A testing x = 0,1,2,3...(Non negative integers)." > > Example: 79 is the least divisor of x^2 + x + 26149427 . > (Its first appearance as minimum divisor) > Ludovicus Huh? 1 is the least (positive) divisor or x^2 + x + 26149427. But perhaps you meant the least prime divisor. Isn't 79 the least prime divisor of x^2 + x + 79? Wait a minute, I think I figured out what you mean: you're saying that no prime less than 79 divides any value of your polynomial. Well, that's no big deal, I think. Suppose I want A such that no number of the form x^2 + x + A is divisible by any prime under 100. For each prime p, x^2 + x takes on about p / 2 different values mod p, so to make sure p doesn't divide x^2 + x + A, I just have to make sure A is in one of the (roughly) p / 2 permitted congruence classes modulo p. Then string them together, using the Chinese Remainder Theorem. This is not so different from the work of Hugh Williams and others, finding numbers N which are not squares mod p for all p up to, say, 300. If 4 c = 1 (mod p) then x^2 + x + A = (x + 2 c)^2 + (A - c), so you're looking for c - A to be a nonsquare mod p. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Rick Decker on 27 Jul 2010 19:50 On 7/27/10 12:21 PM, Ludovicus wrote: > "The set of least divisors of x^2 + x + A is the set of prime > numbers, (A = an odd integer.) > for each A testing x = 0,1,2,3...(Non negative integers)." > This is a bit unclear to me. Let A represent an odd integer, x a nonnegative integer, and p a prime. Do you mean 1. For all A and all p, there is an x such that p is the least divisor of x^2 + x + A. 2. For all p there is an x and an A for which p is the least divisor of x^2 + x + A. 3. Something else entirely. (1) is false: x^2 + x + 3 is never divisible by 7, minimally or not. (2) is true: let x = 1 and A = p^2 - 2 > Example: 79 is the least divisor of x^2 + x + 26149427 . > (Its first appearance as minimum divisor) Oh? What about x^2 + x + 6239 and x = 1? > Ludovicus Regards, Rick
From: sttscitrans on 27 Jul 2010 22:09 On 27 July, 17:21, Ludovicus <luir...(a)yahoo.com> wrote: > "The set of least divisors of x^2 + x + A is the set of prime > numbers, (A = an odd integer.) > for each A testing x = 0,1,2,3...(Non negative integers)." > > Example: 79 is the least divisor of x^2 + x + 26149427 . > (Its first appearance as minimum divisor) > Ludovicus Do you mean that for every A, x^2 + x + A there is a least prime divisor for x=0, 1,2, ... and you are considering the smallest of all the least prime divisors? Is A = 261494427 the first A for which 79 is the smallest prime divisor ? Is A=47 the "first appearance" of 7 as the smallest divisor ?
From: Gerry Myerson on 28 Jul 2010 02:15
In article <1a62f9ae-a1cd-4e61-956e-a1914e940852(a)l14g2000yql.googlegroups.com>, "sttscitrans(a)tesco.net" <sttscitrans(a)tesco.net> wrote: > On 27 July, 17:21, Ludovicus <luir...(a)yahoo.com> wrote: > > "The set of least divisors of �x^2 + x + A is the set of prime > > numbers, (A = an odd integer.) > > for each A �testing �x = 0,1,2,3...(Non negative integers)." > > > > Example: 79 is the least divisor of x^2 + x + 26149427 . > > �(Its �first appearance as minimum divisor) > > Ludovicus > > Do you mean that for every A, x^2 + x + A > there is a least prime divisor for x=0, 1,2, ... and you are > considering the smallest of all the least prime divisors? > Is A = 261494427 the first A for which > 79 is the smallest prime divisor ? > > Is A=47 the "first appearance" of 7 as the > smallest divisor ? x^2 + x is always even, so if A is odd then 2 doesn't divide x^2 + x + A. x^2 + x is always 0 or 2 mod 3, so if A is 2 mod 3 then 3 doesn't divide x^2 + x + A. x^2 + x is 0, 1, or 2 mod 5, so if A is 1 or 2 mod 5 then 5 doesn't divide x^2 + x + A. Putting these together, if A is 11 or 17 mod 30 then 2, 3, and 5 are not divisors of x^2 + x + A. x^2 + x is 0, 2, 5, or 6 mod 7. So A must be 0, 1, 2, or 5 mod 7 for 7 to divide x^2 + x + A. Looking at the numbers that are 11 or 17 mod 30, viz., 11, 17, 41, 47, ..., the first one that is in an admissible class mod 7 is 47. So, yes, it would appear that A = 47 is the first appearance of 7 as the smallest divisor. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |