James Cullen

Quick Info

Born

19 April 1867
Drogheda, County Louth, Ireland

Died

16 December 1933
Clitheroe, Lancashire, England

Summary

James Cullen was an Irish Jesuit and mathematician. He is best known today for his introduction of the Cullen numbers

C_n = n.2^n + 1

and his question about prime Cullen numbers.

Biography

James Cullen was the son of the baker Michael Cullen and his wife Catherine McDonough. He was born at his parents' home (and bakery) at 89 West Street, Drogheda, an industrial and port town on the east coast of Ireland about 40 km north of Dublin. In the 1870s and 1880s West Street had five bakeries, with John Berrill (at No. 53), Thomas Carpenter (at No. 67), Edward Clifford (at No. 88), Michael Cullen (at No. 89), and E & N Moran (at No. 112). West Street, in the centre of Drogheda, runs past St Peter's Parish Church and James was baptised in that church on 23 April 1867 when only four days old.

James' first education was not in a school but he was educated privately. The first school he attended was run by the Christian Brothers. The Congregation of Christian Brothers is a religious community within the Roman Catholic Church founded around 1820. The Christian Brothers first arrived in Drogheda in 1857 and the foundation stone for the King Street monastery and school complex was laid on 19 March 1867 by the Most Rev Dr Kieran. Cullen excelled at this school showing particular abilities in mathematics. He graduated from the Christian Brothers school in Drogheda and matriculated at Trinity College, Dublin to study Pure and Applied Mathematics.

After graduating from Trinity College, Dublin, he was in business for a while, but in May 1891, at the age of 24, he decided that he wanted to become a priest. His training had been in mathematics and he had not studied the Latin which would be necessary for the priesthood in the Catholic Church. He therefore went Mungret College situated about five km west of Limerick City, which had been a Jesuit apostolic school from 1882. After studying Latin at Mungret College, in 1892 he entered the noviceship of the English Jesuits at Manresa House, Roehampton, London.

Manresa House had been originally Parkstead but in 1861 it had been sold to the Society of Jesus who used the house as a novitiate. They renamed the house Manresa House after the town in Spain where Ignatius of Loyola, the founder of the Jesuits, worked out the principles that the Jesuits would follow. From 1892 till 1895 Cullen was at Manresa training for membership of the Society of Jesus after which he remained there for a further year's study. He then studied philosophy at St Mary's Hall, Stonyhurst from 1895 till 1897. St Mary's Hall was, at that time, a place for philosophical studies and a training seminary for the British Province of the Society of Jesus. Located in Lancashire, England, the building had been constructed between 1828 and 1830. It served as a critical intellectual hub for young Jesuits undergoing their rigorous priestly formation. From 1897 till 1901 Cullen studied theology at St Beuno's College in North Wales. Built in 1847 by the Jesuits as a theology college, Beuno's College was situated on the Clwydian Range overlooking the Vale of Clwyd with views facing west toward Snowdonia. Cullen was ordained priest at St Beuno's College on 31 July 1901.

Although he had spent years training for the priesthood, Cullen had continued his interest in mathematics and he was particularly fascinated by number theory. After his ordination, he taught mathematics at Manresa House, in Roehampton, to young Jesuits who had finished their noviceship. He was also undertaking research and publishing his results in high quality journals. He submitted the paper The Solutions of a System of Linear Congruences to the Proceedings of the London Mathematical Society in March 1902 (as stated on the published paper). Cullen headed the paper with the following Acknowledgement [3]:-

The author is indebted to Lt Col Allan Cunningham, R.E., not only for having carefully revised the original draft, but also for many important suggestions embodied in the paper.

We note that Lt Col Allan Joseph Champneys Cunningham, R.E. (1842-1928) was born to British parents in India and had a military career in India before returning to the U.K. in 1881 and taught mathematics at various military institutions. After leaving the army in 1891 he spent the rest of his life studying number theory. He continued to be involved in research relating to ideas introduced by Cullen as we shall see below.

Returning to Cullen's paper The Solutions of a System of Linear Congruences we quote from the Introduction [3]:-

The object of the present paper is to give a graphical process for obtaining solutions satisfying a system of linear congruences within a given limit. The process is adapted to the factorisation of large composites, or the determination of large primes and other problems in the theory of numbers, such, for instance, as the representation of high numbers in binary quadratic forms, &c.

Incidentally some properties of a system of linear congruences are given that are believed to be new.

The scope of the paper consists in proving and explaining four simple rules to be employed in the application of the process, together with an example showing its working.

In April 1904 he submitted the follow-up paper [4] to the Proceedings of the London Mathematical Society. It begins as follows:-

Gauss (Werke, Vol. 1., pp. 507-509) has given a table from which the linear forms of H in the equation

$N = \ell H^{2} + mG^{2}$

are easily obtained. In a previous communication ([3] in our references) the writer has given a general process for solving any linear system. When, however, H is ambiguous in sign the work in applying this process can be considerably reduced. The object of the present note is to explain this briefly.

In 1905 Cullen published [5], namely Question 15897 in the Educational Times. It is this, apparently very minor, piece of mathematics for which today James Cullen is best remembered. For example, looking at MathSciNet in September 2026 we see that there are 24 papers with Cullen's name in the title referring to ideas which originate in Question 15897. Cullen defined, what today are called the Cullen numbers, as follows

C_{n} = n.2^{n} + 1

He showed that for

1 < n < 100, C_{n}

was composite except for possibly

n = 53

. His question was simply, was

C_{53}

prime or composite?

The question was quickly solved, Allan J C Cunningham, whom we mentioned above, showed that

C_{53}

was composite, showing that it is divisible by 5591 (see [10]). In fact it is easy to answer questions like that today. We easily calculate the prime decomposition

C53 = 477,381,560,501,272,577 = 5591 × 16747 × 5098461301.

In [10] Cunningham showed that for

1 < n ≤ 200, C_{n}

was composite except for possibly

n = 141

. He remarked that primes of the form

C_{n} = n.2^{n} + 1

seem to be remarkably rare. His question regarding whether

C_{141}

was prime or composite took 50 years to answer.

We give a little more information below about how studying the Cullen numbers developed but first let us finish Cullen's biography.

In 1905 he taught Mathematics at the Jesuit boarding school Mount St Mary's College, Spinkhill in Derbyshire. T Geoffrey Holt, an archivist at the Department of Historiography and Archives of the English Province of the Society of Jesus, writes about Cullen in [12]:-

It is said that he found difficulty in teaching.

The result seems to be that Cullen stopped being a mathematics teacher and worked for the Jesuits as an accountant. In 1906 he was sent to Stonyhurst College, where he had studied philosophy 1895-1897. Here he served as an accountant and after about seven years he became Procurator of the College, the farm and estate. The College went through a difficult period during World War I and closed its undergraduate philosophy programme in 1916.

Although he had stopped teaching mathematics, Cullen kept in touch with leading mathematicians and continued to contributed articles to Nature, the Mathematical Gazette and the Messenger of Mathematics. In 1921 he left Stonyhurst and was appointed to supervise accounts of other English Jesuit centres such as: Wimbledon College on Edge Hill, Wimbledon; the Sacred Heart Parish in Blackpool; and St Wilfrid's Parish in Preston. His health deteriorated and he took leave for a period to try to recover. In November 1933 he returned to his duties in Wilfrid's Parish but in December went to Stonyhurst College for his annual retreat [16]:-

While making a stay at Stonyhurst College for his annual retreat, Father James Cullen, S.J., was seized, on the 6th December, with sudden illness. He was engaged, at the time, in offering the Holy Sacrifice, and was able to finish Mass only with the utmost difficulty. The state of his heart prevented any treatment by way of operation, and early on the following day, after receiving Extreme Unction, the stricken priest died.

A requiem Mass for Father James Cullen S.J. was celebrated at Stonyhurst on Monday, 11 December 1933, followed by interment in the private cemetery of Stonyhurst College.

Let us now give a few details of how the study of the Cullen numbers developed.

We left our description of Cullen numbers with Allan Cunningham answering Cullen question of whether C_{53} was prime by showing it was composite. Cunningham posed the question in 1910 whether

C_{141}

was prime and this was not answered until 1958 when Raphael Robinson proved that indeed

C_{141}

is prime. If you are wondering exactly how big a number

C_{141}

is then we can give it

C_{141}

= 393050634124102232869567034555427371542904833.

Allan Cunningham and Herbert J Woodall, intrigued by Cullen's question, published the paper [9] in 1917 in which they investigated factorising both Cullen numbers and the related sequence

W_{n} = n.2^{n} - 1

now sometimes called Woodall numbers, sometimes called Cullen numbers of the second kind. They had been introduced by Woodall in a 1911 paper.

In 1995 Wilfred Keller [14] showed that the Cullen number

C_{n}

was prime for

n = 4713, 5795, 6611, 18496

. Since Keller's paper,

C_{n}

has been shown to be prime for

n =

32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881. The last two in this list were found in 2009 and, as far as we know, no further Cullen primes have been found (up to 2026). It was shown in 2015 that there are no other Cullen primes for

n < 13705481

. Many questions remain open, for example:

Are there infinitely many Cullen primes?

Does there exist a prime

p

such that

C_{p}

is prime?

At present (2026) 34 of the numbers

W_{n}

have been shown to be prime, the largest being

W_{n}

for

n = 17016602

.

The concept of a Cullen number has been generalised to

s

-Cullen numbers

C_{s,n} = n.s^{n}+ 1

It has been proved by Christopher Hooley in 1976 that 'almost all' Cullen numbers are composite. In 2004 it was shown that the intersection of the Cullen numbers with the Fibonacci sequence is finite. In 2014 it was shown that the intersection of the

s

-Cullen numbers with the Fibonacci sequence is finite. Several papers (for example [11]) look at the intersection of the Cullen numbers with other recurrent sequences.

Other Mathematicians born in Ireland

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Written by J J O'Connor and E F Robertson
Last Update October 2026