# Which
came first: 180(2^{127}-1)^{2} + 1 or (2^{148}-1)/17?

### By Chris Caldwell

## The Problem:

For seventy five years Lucas' Mersenne:
M_{127} = 2^{127}-1 held the record for the largest known
prime. In 1951 Miller and Wheeler beat this by finding a series
of primes *k**M_{127}+1, then 180(2^{127}-1)^{2} +
1. The same year Ferrier
announced (2^{148}-1)/17. So which came first? This
web page addresses just this one point of contention as a supplement
to our document: The
Largest Known Prime by Year.

## Answer One: I don't know.

In the journal Nature, Miller and Wheeler [MW51] wrote the following (this is the complete text of the article):

Large Prime Numbers

For about seventy-five years the largest known
prime number has been *P* = 2^{127} - 1,
identified as such by Edouard Lucas. It has remained the largest
known in spite of many attempts to identify larger
ones, although there have been conjectures and claims
insufficiently substantiated or entirely unproved.

Recently we have prepared a routine for testing on the
EDSAC the primality of numbers of the form *k P* + 1, with *P* as
defined above, and have found ten values of *k* that give prime numbers.
The largest of these gives the present (June 7) largest known prime, namely,

934(2^{127} - 1) + 1

J. C. P. Miller

D. J. Miller

June 7.

*Note added in proof (October 8)*. Further work on
the EDSAC by Wheeler and myself has demonstrated the primality of

978(2^{127} - 1) + 1

and culminated in early July in the identification of the present largest known prime,

^{127}- 1 )

^{2}+ 1.

^{148}+ 1)/17,

University Mathematical Laboratory, Cambridge.

This can be read in many ways. If Miller & Wheeler knew of Ferrier's work in June, this would seem to imply Ferrier had not finished when they found their first sequence of primes, but even so, it does not order the two "early July primes." So possibly Miller & Wheeler had a sequence of records sized primes beginning in June:

*k*

^{.}M

_{127}+ 1 for

*k*= 114, 124, 388, 408, 498, 696, 738, 774, 780, 934 and 978;

^{127}-1 )

^{2}+ 1. In this case Ferrier's (2

^{148}+1)/17 never held the record for the largest known prime.

Or possibly Miller & Wheeler had a sequence of records sized primes beginning
in June and then discovered Ferrier's (2^{148}+1)/17 far surpassed
them. Then after that they searched for a different form, one to surpass
Ferrier, finding 180(2^{127}-1
)^{2} +
1. Otherwise, why did they change in form? And why find just one
of them? (They
would have needed to go beyond *k*=123362 to surpass Ferrier with the
form *k*^{.}M_{127}+1,
and the next prime of the second form is 330(2^{127}-1)^{2} +
1.)

I checked with a family member, David Miller, who responded [e-mail, 4 April 1997]:

I do not know which was first. I have just checked some notes of J. C. P. Miller in which he states "until the 8th was found to be prime in July." and "during the period of May to July A. Ferrier was applying similar tests to (2^{148}+1)/17, using a desk machine. His results show this to be prime, the second largest prime number known." The notes are undated so I cannot deduce which was first. My failing memory recalls that Jeff Miller, went to some length to make sure Ferrier's result was not overlooked. So I would guess that the 180 was first with a probability of about 2/3rds.

Undated... darn. So what do we know of Ferrier's dates? We definitely know he had completed his work by July 14, but as Hans Riesel suggest, he may have been done earlier [e-mail, 26 March 1997]:

I saw your "history question" on the largest prime, established without computers. In MTAC, the forerunner to Math. Comp, vol. 6 (1952). p. 256, Ferrier reports on his work on (2^{148}+1)/17. There is an editorial remark, stating that Ferrier's letter to MTAC is dated July 14, 1951. So at least at that date Ferrier had completed his primality proof for this number.

(Perhaps Ferrier already had it for some time, and as a good Frenchman saved its announcement to the French national holiday, but this is just a guess on my part.) A description of Ferrier's work on this number can be found in my book, on pp. 122-123. So, if you cannot find a more precise date for Wheeler's and Miller's work, it will be difficult to tell, which number came first.

If indeed he had waited for Bastille
Day, July 14th, he surely would
not have waited too long. The one page excerpt of Ferrier's letter
in MTAC is a very nice summary, but gives no hint on timing. It
should not have taken months, so if he had been working on it in May,
he *could* have
completed before July. (Just like with today's prime records,
it is not the test of the final number that takes so long--it is finding
the right number to test!)

It is intriguing to me that D. H. Lehmer listed Ferrier first when summarizing "recent discoveries" (given in its entirety) [MTAC, Vol. 5, No. 36, Oct., 1951] :

131. Recent Discoveries of Large Primes. Ever since Lucas announced the discovery of the prime 2^{127}-1 in 1786, many attempts have been made to discover larger primes. These attempts have succeeded only recently as follows:

(a) A. Ferrier1 has identified (2^{148}+1)/17 as prime, using a method based on the converse of Fermat's theorem and a desk calculator.

(b) Using the same method and the EDASC, Wheeler and Miller2,3 have proved the primality of 1+k(2^{127}- 1) fork= 114, 124, 388, 408, 498,696, 738, 744[sic], 780, 934, 978, and finally 1+180(2^{127}- 1)^{2}, a number of 79 decimal digits.

(c) Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2^{521}-1 and 2^{607}-1 on January 30, 1952. These led to the 13th and 14th perfect numbers.D. H. L.

^{1}Letter of July 14, 1951.

^{2}J.C.P. Miller & D. J. Wheeler, "Large prime numbers,"Nature, v. 168, 1951, p. 838.

^{3}J.C.P. Miller, "Large Primes,"Eureka, 1951, no. 14, p. 10-11.

Again, there is no statement that the order is chronological.

## Answer Two: Maybe it Doesn't Matter

The two records are of very different type. Whichever was first, Ferrier's result stands as the largest known prime found manually. On the other hand, Miller and Wheeler's record lasted only a matter of months.

On our page of the Largest Known Primes by Year, the two records stand in separate sections--Ferrier's at the end of the "before computer" primes and Miller and Wheeler began the computer era of discovery. Both records are important in their own right, and though I would be interested in knowing the exact dates of discovery (let me know if you have information!), I think we can rest for now with Lehmer's summary.