Conference rusure::math

The following reply is a SIXEL image of the resulting pattern.
It has the proper introducer and terminator sequences to allow it to be
TYPEd out after SAVing.  The line counter in NOTES-11 turns off SIXEL
mode (as I have my terminal line set up, anyway), so the image will not be
completely displayed if you just READ it.

ADVISORY NOTE:
The following reply may cause grief to non-graphics terminal users.
It is suggested that you skip the next reply!

- tom]

When including a picture (or other fancy graphics), please give a short
description and/or warning, followed by a form feed, then the picture.
That way, the innocent users have the opportunity to avoid it, and the
intended audience can "SAVE TT:" to see it, without first getting stuck
in a funny video mode.

*** WARNING -- The following contains VT125 graphics ***

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@!5?A!13?C?A!5?@!17?A-
-
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\

C
C	Here's a version where the primes of the form 6k+1 and the primes
C	of the form 6k-1 are colored with two different colors (red and green).
C
	PROGRAM PRIMEPLOTCOLOR
	IMPLICIT INTEGER*4(A-Z)
	PARAMETER (TOP=57600)
	LOGICAL*1 PRIMES(TOP)
C
C	Plots primes in a spiral on a VT125.
C
C	Stanley Rabinowitz
C
C	The numbers from 1 to 57600 are drawn in a spiral as follows:
C
C
C              17  16  15  14  13
C              18   5   4   3  12
C	       19   6   1   2  11
C              20   7   8   9  10
C              21  22  23  24  25
C
C	Each prime is displayed as a dot; composite numbers are not displayed.
C	Interesting unexpected patterns are seen.
C	Primes of the form 6k+1 are colored differently from primes of
C	the form 6k-1.
C
	BYTE DELTAX(4) /+1, 0,-1, 0/
	BYTE DELTAY(4) / 0,+1, 0,-1/
	CALL SIEVE(PRIMES,TOP)
	CALL CLEAR
	X=0
	Y=0
	POS=1
	NUMBER=1
	DO 2 K=1,TOP
C
C	Right k, Up k, Left k+1, Down k+1, etc.
C
	DO 2 JJ=1,2
	DO 3 J=1,K
	IF (NUMBER.EQ.TOP) GOTO 50
	NUMBER=NUMBER+1
	X=X+DELTAX(POS)
	Y=Y+DELTAY(POS)
	IF (PRIMES(NUMBER)) THEN
		REM=MOD(NUMBER,6)
		IF (REM.EQ.1) THEN
			CALL PLOT(X,Y,'R')
		ELSE
			CALL PLOT(X,Y,'G')
		END IF
	END IF
3	CONTINUE
	POS=POS+1
	IF (POS.EQ.5) POS=1
2	CONTINUE
50	TYPE 100, 27
100	FORMAT(1X,A1,'\')
	CALL EXIT
	END

	SUBROUTINE SIEVE(PRIMES,TOP)
	IMPLICIT INTEGER*4(A-Z)
	LOGICAL*1 PRIMES(TOP)
C
C	Calculate primes to 40,000 using Sieve of Eratosthenes.
C	For purposes of this program, treat 1 as a prime.
C
	INTEGER*2 PR(7)
	DATA PR/3,5,7,11,13,17,19/
	DO I=2,TOP,2
	PRIMES(I-1)=.TRUE.
	PRIMES(I)=.FALSE.
	END DO
	DO 2 I=1,7
	P=PR(I)
	DO 3 J=3*P,TOP,2*P
3	PRIMES(J)=.FALSE.
2	CONTINUE
	DO 4 I=3,300,2
	IF (.NOT.PRIMES(I)) GOTO 4
	DO 5 J=3*I,TOP,2*I
5	PRIMES(J)=.FALSE.
4	CONTINUE
	RETURN
	END

	SUBROUTINE CLEAR
	IMPLICIT INTEGER*4(A-Z)
	ESC=27
	TYPE 100, ESC,ESC,ESC
100	FORMAT(1X,A1,'[H',A1,'[2J',A1,'P1ps(M0(L0)(AL0))S(I0)S(E)S[0,0]')
	RETURN
	END

	SUBROUTINE PLOT(X,Y,COLOR)
	IMPLICIT INTEGER*4(A-Z)
	PARAMETER (XOFFSET=350)
	PARAMETER (YOFFSET=240)
	CHARACTER*1 PREV_COLOR/'X'/
	CHARACTER*1 COLOR
	IF (COLOR.NE.PREV_COLOR) THEN
		TYPE 101, COLOR
101		FORMAT('+W(I(',A1,$,'))')
		PREV_COLOR=COLOR
	END IF
	TYPE 100, 2*X+XOFFSET,480-2*Y-YOFFSET
100	FORMAT(' P[',I3,',',I3,']V[]')
	RETURN
	END

Path: ryn.esg.dec.com!shlump.nac.dec.com!bacchus.pa.dec.com!decuac!haven!cvl!herve
From: [email protected] (Jean-Yves Herve')
Newsgroups: sci.math
Subject: Re: Pictures of primes in the plane
Keywords: Prime numbers
Date: 1 Oct 90 22:08:00 GMT
References: <[email protected]>
Reply-To: [email protected] (Jean-Yves Herve')
Organization: Center for Automation Research, Univ. of Md.
 
In article <[email protected]> [email protected] (Cleve Moler) writes:
>When we were both graduate students in early '60s, Larry Breed
>showed me a fascinating image he had generated that displayed
>some interesting properties of prime numbers.  Since then, I've
... 
>Number the points in the plane with integer coordinates in a
>"square spiral" pattern as follows:
... 
>Whose idea was this?
 
>   -- Cleve Moler
>   [email protected]
 
I remember seing this pattern called the Ulam Spiral (as in Stanislaw M. Ulam).
Don't know of any reference, though.
 
Jean-Yves.
 
-------------------------------------------------------------------
 Jean-Yves Herve'             
 Computer Vision Laboratory		[email protected]
 Center for Automation Research           
 University of Maryland
-------------------------------------------------------------------