In his book Micro-Maths, ISBN 0-333-39007-5, Macmillan 1984, Keith
Devlin provides the polynomial in question. I will check my typing,
but I am almost certain I will err. ^2 will mean squared, of course.
The variables, not surprisingly, are a to z.
(k + 2) { 1 - [wz + h + j - q]^2 - [(gk + 2g + k + 1).
(h + j) + h - z]^2 - [2n + p + q + z - e]^2
- [16(k + 1)^3 . (k + 2) . (n + 1)^2 + 1 - f^2]^2
- [e^3 . (e + 2) (a + 1)^2 + 1 - o^2]^2 - [(a^2 - 1)y^2
+ 1 - x^2]^2 - [16r^2y^4(a^2 - 1) + 1 - u^2]^2
- [((a + u^2(u^2 - a))^2 - 1) . (n + 4dy)^2 + 1
- (x + cu)^2 ]^2 - [n + l + v - y]^2 - [(a^2 - 1)l^2 + 1 - m^2]^2
- [ai + k + 1 - l - i]^2 - [p + l(a - n - 1)
+ b(2an + 2a - n^2 - 2n - 2) - m]^2 - [q + y(a - p - 1)
+ s(2ap + 2a - p^2 - 2p - 2) - x]^2 - [z + pl(a - p)
+ t(2ap - p^2 - 1) - pm]^2 }
Substitute non-negative whole numbers for each variable a to z and
positive results are primes. Formula due to Jones, Sato, Wada and
Wiens, 1977 after Davies, Matijasevic, Putnam and Robinson had proved
such a formula has to exist.
Devlin is a mathematician who popularised mathematics in his
long-running column in the British newspaper "The Guardian". The first
chapter of Micro-Maths is called "Computer mathematics reaches its
prime and tells of the use of computers to find Mersenne primes up to
M29 in September 1983. "Record primes have little interest for the
professional mathematician, but they certainly have a habit of hitting
the newspaper and TV screens." Nevertheless, his next book "Microchip
Mathematics", Shiva 1984, which is far less accessible to the layman
than "Micro-Maths" carries a picture of Devlin holding up a printout
of M29, which according to notes took over half an hour of mainframe
computer time to calculate.
Regards,
Ian
On 24 Jan 2006 08:44:10 +0000, Paul Leyland <[EMAIL PROTECTED]> wrote:
> On Tue, 2006-01-24 at 03:37, Mike McCarty wrote:
>
> > ISTR that there is a polynomial in 26 variables which, when the
> > various variables take on all integral values, generates only
> > primes (and 1, IIRC).
>
> You remember incorrectly.
>
> The polynomial in question generates an infinite number of composite
> numbers, all negative. Its *positive* values are prime.
>
>
> Paul
>
>
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--
Ian W Halliday, BA Hons, SA Fin, ATMG, CL
+44 772 546 2965 (GMT)
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