# Ten Primes

A search for ten consecutive primes in arithmetic progression

# Success!!!

### On 2 March 1998 using my PC/Windows program, CP10, Manfred Toplic found TEN CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSION

The primes are

P = 100 99697 24697 14247 63778 66555 87969 84032 95093 24689 19004 18036 03417 75890 43417 03348 88215 90672 29719,

P + 210, P + 420, P + 630, P + 840, P + 1050, P + 1260, P + 1470, P + 1680 and P + 1890.

## Background

On August 29 1995, Harvey Dubner and Harry Nelson discovered seven consecutive primes in arithmetic progression, namely p, p + 210, p + 420, p + 630, p + 840, p + 1050 and p + 1260 where p is the 97-digit number:
1089533431247059310875780378922957732908036492993138195385213105561742150447308967213141717486151.

On 7 November 1997, we - Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann - announced the discovery of eight consecutive primes in arithmetic progression after a search lasting a couple of months using a variety of PC's and Unix workstations.

We then initiated a search for nine primes, calling on the help of about a hundred people world-wide and using about 200 computers. On 15 January 1998, this search ended successfully when Manfred Toplic of Klagenfurt, Austria, found nine consecutive primes in arithmetic progression using Tony's program CP09.EXE on a PC.

## Ten Primes

Immediately we started to look for ten consecutive primes in arithmetic progression. Although much more difficult, it was nevertheless quite feasible. We estimated about 300 years, assuming computers search at the rate of 1200 million numbers per hour. Therefore we thought we would need a lot of help.

We looked for special candidates of the form p = N m + x, where

x = 54538241683887582668189703590110659057865934764604873840781923513421103495579,

m = 193#, the product of the 44 primes up to 193, and N = 0, 1, 2, ....

We think we were unlucky with nine primes; the search took considerably longer than we anticipated. The probability calculation for ten primes suggests that we should expect success after about 3000 trillion Ns, give or take a quadrillion or two. Well, as you have seen, this time we good fortune. The result came after only about 48 trillion numbers were tested when the same Manfred Toplic found the winning N.

## Further Information

### TEN PRIMES

Newsletter/3 - the official announcement : 7 March 1998.

Ivars Peterson's report in Math Trek : 23 March 1998.

The paper : 28 November 2001.

Graphical representation of 214 primes in the vicinity of 100 99697 24697 14247 63778 66555 87969 84032 95093 24689 19004 18036 03417 75890 43417 03348 88215 90672 29719

### NINE PRIMES

Keith Devlin's article in The Guardian : 19th February 1998.

### References

Harvey Dubner and Harry Nelson, Seven primes in arithmetic progression, Mathematics of Computation, October 1997. Also available as a PDF file.

Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann, 8 consec. primes in AP, NMBRTHRY, November 1997.

Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann, 9 consecutive primes in arithmetic progression, NMBRTHRY, January 1998.

Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony, H. Nelson and Paul Zimmermann, Ten consecutive primes in arithmetic progression, Mathematics of Computation 71 (2002). Also available as a PDF file.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, 2nd edition, Section A.6.

Ivars Peterson, A Progression of Primes, Ivars Peterson's MathTrek, 17 November 1997.

Ivars Peterson, Nine Primes in a Row, Ivars Peterson's MathTrek, 9 February 1998.

Ivars Peterson, A Prime Surprise, Ivars Peterson's MathTrek, 23 March 1998.

Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1995, 3rd edition, Section IV.C.