## Bronowski and retrodigitisation

**Jacob Bronowski (1908--1974),**

*The Mathematical Gazette*and retrodigitisation
**by D G Rogers**

Peter Braza and Tong Jing-Cheng, in an article [*The Mathematical Gazette* in 1999, considered moving the leading digit of an integer to the rear so as to form a new integer that is a multiple of the original. Subsequently, Jeremy King wrote in [*Gazette* in the early 1950s that still merits recalling, although it might also be noted that Bob Burn had outlined an approach in another article [

The issue of *The New Statesman* for 24 December, 1949 (p. 761), included a challenge from Jacob Bronowski (1908--1974): find the least integer (in base 10) such that moving the leading digit to the rear produces a new integer one and a half times the original. This puzzle carried the warning that computation might prove lengthy, and, indeed, the answer runs to 16 digits. While no source was given, such problems had long been popular; for example, Maurice Kraitchik (1882--1957) poses several in which the first or final digit and the multiplier are specified in his popular book *Récreations Mathématiques* [*A History of the Theory of Numbers* [

Bronowski's problem was given an airing [*Gazette* in 1952 by J H Clarke, whose approach was somewhat along the lines adopted more recently in [*Gazette,* R L Goodstein (1912--1985), who noted [*Gazette,* when Lt.-Col. Allan J C Cunningham (1842--1928) published tables [

In these latter days of Research Assessment Exercises, it is perhaps worth remembering that Littlewood, Primrose and Goodstein were academic mathematicians of some distinction (see the obituary notices [*Gazette* throughout a career spent entirely at University College, Leicester (later University of Leicester), from 1947 until 1982, during which time he showed great deftness in the construction of geometrical designs. Louis Goodstein was Primrose's professorial colleague at Leicester from 1948 until retirement in 1977, being the first specialist in mathematical logic to hold a mathematical chair in the UK. He published prolifically in the *Gazette,* serving as Editor in the period 1956--1962. Naturally, his Presidential Address [*Gazette* later that year. On the other hand, although Robert Sibson, like Goodstein a year earlier, was a Wrangler in the Mathematical Tripos at Magdalene College, Cambridge in 1934, he pursued a different career, going into school teaching, and thence joining HM Inspectorate of Schools in 1947, retiring in 1973. His final appointment was as Joint Secretary of the Schools Council.

Although Jacob Bronowski's name is most remembered in association with the BBC television documentary series The Ascent of Man he made at the end of his life - it inspired discussion [*Gazette* of a tessellation found at the Alhambra - he read mathematics at the University of Cambridge, where he went on to take a doctorate with a thesis in geometry and topology. It would seem that he distilled something of his own experience in the gleaning [*Gazette* in 1953, but which retains a certain quotable resonance today:

However, he does not seem to have published anything in theThe scientist, pure or applied, is still often treated as an uncouth Philistine. I find this a perverse charge, coming from men whose cultural interests fit primly into the clues to The Times crossword.

*Gazette*until 1963, when he was at the National Coal Board. But appropriately enough, his note [

*Gazette,*too numerous to detail here.

Other observations on digits have appeared in the *Gazette* from time to time; for example, on the cycles of integers that can be formed on repeatedly taking certain sums of consecutive digits [*Disquisitiones Arithmeticae,* raised the question of determining the primes p for which 10 is a primitive root modulo p. Emil Artin (1898--1962) distilled the ensuing investigations of special cases in a general conjecture in 1927: any integer m, other than 0 or -1, and not divisible by a square, is the primitive root of infinitely many primes; and such primes have positive denisity in the set of primes independent of the choice of m. Although much progress has been made on this conjecture, it has been of a conditional or non-constructive kind, and, as yet, no m is known which is a primitive root for infinitely many primes.

**References**

[*Math. Gaz.* **83** (1999), 216--220.

[*Math. Gaz.* **84** (2000), 125.

[*Math. Gaz.* **75** (1991), 154--157.

[*La mathématique des jeux ou Récreations Mathématiques* (Steven Frères, Brussels, 1930; 2nd. ed., Editions techniques et scientifiques, Brussels, 1953); in English trans. as (b) *Mathematical Recreations* (W W Norton, New York, NY, 1942; George Allen and Unwin, London, UK, 1943; 2nd. ed., Dover Pub. Inc., New York, NY, 1953).

[*A History of the Theory of Numbers* Vol I (Carnegie Institute, Washington, DC, 1919; reprinted Chelsea Pub., New York, NY, 1952; Dover Pub. Inc., New York, NY, 2005).

[*A Treatise on the Elements of Algebra* (H Ingham, London, 1854).

[*Math. Gaz.* **36** (1952), 276.

[*Math. Gaz.* **39** (1955), 58.

[*Math. Gaz.* **39** (1955), 58--59.

[*Math. Gaz.* **39** (1955), 59.

[*Math. Gaz.* **40** (1956), 131--132.

[*Math. Gaz.* **4** (1907--1908), 259--267; (b) On tertial, quintic, etc. fractions, *ibid* **6** (1911--1912), 63--67; (c) On tertial, quintic, etc. fractions (continued), *ibid* **6** (1911--1912), 108--116.

[*Bull. London. Math. Soc.* **15** (1983), 56--69.

[*Bull. London Math. Soc.* **34** (2002), 495--501.

[*Bull. London Math. Soc.* **20** (1988), 159--166.

[*Math. Gaz.* **60** (1976), 165--170.

[*Math. Gaz.* **73** (1989), 297--301.

[*Math. Gaz.* **74** (1990), 372-- 373.

[*The Observer,* 22 April, 1951, per Mr P Vermes, *Math. Gaz.* **37** (1953), 89.

[*Math. Gaz.* **47** (1963), 234--235.

[*Math. Gaz.* **25** (1941), 156--159; (b) A digit transformation, *ibid* **40** (1956), 20--21.

[*p*^{e}), *Math. Gaz.* **59** (1975), 195.

JOC/EFR August 2007

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