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The Arora-Barak book on complexity contains the following sentence with the following footnote, page 128:

In primality testing, we are given an integer N and wish to determine whether or not it is prime. Algorithms for primality testing were sought after even before the advent of computers, as mathematicians needed them to test various conjectures [3].

[3]: An interesting anecdote is that Gauss, even though he was a very fast human computer himself, used the help of a human supercomputer—an autistic person who excelled at fast calculations—to do primality testing.

I would like to read more about this, but I cannot find anything close to a source on this, or even a repetition of this.

I am not looking for a relationship between autistic people and prime numbers, or mathematics, but specifically that Gauss had an autistic assistant he tasked with checking primality.

My question is that I am asking for confirmation if this is even true, and if so, to provide further references.

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    $\begingroup$ My guess is that you are refering to the mental calculator Zacharias Dase. $\endgroup$
    – user2554
    Jul 31, 2023 at 20:37
  • $\begingroup$ @user2554 Thanks this gives me a place to begin searching. $\endgroup$ Jul 31, 2023 at 21:35
  • $\begingroup$ Your choice of vocabulary is unsettling. Comparing humans with computers is one thing. Comparing humans with computers in conjunction with applying the word "use" to these humans appears to objectifies them and is pretty distasteful. $\endgroup$
    – Stef
    Aug 2, 2023 at 12:47
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    $\begingroup$ @stef no problem with your sentiment, but the OP is quoting a book, so not sure why you are accusing them personally. Furthermore, in Gauss' day a computer was a profession, so there is no comparison nor metaphor here. $\endgroup$
    – Yorik
    Aug 2, 2023 at 16:29
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    $\begingroup$ @Yorik The book says "used the help of a human supercomputer", i.e. the object of the verb "used" is the noun "help", not the person. In the question title, the object of the verb "used" is the autistic assistant. Using somebody's help is rather different to using somebody, at least in connotation. $\endgroup$
    – kaya3
    Aug 3, 2023 at 13:31

1 Answer 1

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The human computer was the computational prodigy Johann Martin Zacharias Dase (June 23, 1824 - September 11, 1861), who was born in Hamburg, Germany and also died there. He reportedly attended school from the age of $2\frac{1}{2}$, and at the age of $15$ began to travel (initially in Northern Germany) to make a meager living by exhibiting extraordinary feats of mental calculation. It appears that he never held a regular job for the duration of his life. His travels eventually brought him to Vienna, and his stay there led to his first publication in Crelle's Journal, on the computation of $\pi$ to 200 decimal places:

Z. Dahse, "Der Kreis-Umfang für den Durchmesser 1 auf 200 Decimalstellen berechnet", Journal für die reine und angewandte Mathematik, Vol. 27, No. 3, 1844, p. 198 (scan)

This result was submitted to the journal by a Viennese professor von Schulz Strasznicky, who reports that Dase had arrived in Vienna in 1840 and had to rely on the kind support of various benefactors, among them the Scots Abbey. The professor had asked Dase to put his computational abilities to work on a scientifically useful result by calculating $\pi$ to 200 decimals. Dase chose the formula $\frac{1}{4} \pi = \mathrm{arc tang} \frac{1}{2} + \mathrm{arc tang} \frac{1}{8} + \mathrm{arc tang} \frac{1}{5}$ for this work. von Schulz Strasznicky further reports that Dase had found temporary employment in the head office of the state railways and had commenced work on a table of natural logarithms with seven decimal places.

One of Gauss's regular correspondents, H. C. Schumacher from Altona near Hamburg (today a borough of Hamburg), wrote in a letter to Gauss dated January 1, 1847 that Dase has returned to Hamburg and that two-thirds of the table of logarithms are complete, but that he anticipates Dase will have problems trying to get it published. He further mentions that he found a discrepancy between Dase's value of $\pi$ and the earlier result of Rutherford after the 152nd digit:

C.A.F. Peters (ed.), Schriftwechsel zwischen C. F. Gauss und H. C. Schumacher, 5. Band, Altona: G. Esch 1863, "No. 1131 Schumacher an Gauss", pp. 275-279 (scan)

Ein junger Mensch, Dase, von dem ich Ihnen schon vor einigen Jahren schrieb, und der Reisen gemacht hat, um Vorstellungen seines Talentes im Kopfe zu rechnen zu geben, ist nun wieder hier und ein merkwürdiges Beispiel, wie weit Intuition von Zahlenverhältnissen gebracht werden kann.
[...]
Er hat jetzt die natürlichen (7stelligen) Logarithmen der Zahlen von 1 bis 1005000 berechnet, wovon $\frac{2}{3}$ schon fertig sind, und der Rest vor Ostern fertig wird, und sucht einen Verleger, den er aber schwerlich finden wird. $\pi$ hat er bis zu 200 Decimalen berechnet. Ich habe seine Zahlen mit denen von Rutherford (Phil . Tr. 1841. Pt. II. p. 283) verglichen und finde
1) bis zur 152. Decimale incl. stimmen sie genau

After consulting with Thomas Clausen, Schumacher informed Gauss in a letter dated January 22, 1847 that Dase's result for $\pi$ is entirely correct. In correspondence over the next few months, Schumacher and Gauss discussed in more detail Dase's capabilities and how one might put them to use for scientific projects. In a letter dated April 18, 1847 Schumacher informs Gauss that Dase is currently employed in Prussia, but that he is still in a probationary period and that he (contrary to Dase), does not expect him to be employed there permanently. He mentions that Dase has had plans to extend the table of factors to 10 million for quite a while and that the academy in Berlin might take him up on that should they continue to pay him a salary:

In Preussen ist er noch in einer Probe-Anstellung. Der Termin läuft aber diesen Sommer ab, und ich möchte nach Encke's Aeusserungen bezeifeln, dass man ihn ausdehnen werde, was er selbst ziemlich sicher zu erwarten scheint. Die Factorentafel bis auf 10 Mill. auszudehnen, ist schon lange sein Plan gewesen und dazu, scheint es mir, könnte die Berliner Akademie ihn gebrauchen, wenn man ihm sonst noch ferner Gehalt geben will

Schumacher's prediction about the difficulty of finding a publisher for Dase's table of logarithms appears to have been accurate, because it took another three years before this was published:

Zacharias Dase, Tafel der natürlichen Logarithmen der Zahlen. Vienna: Leopold Sommer 1850 (Google scan)

In the preface, Dase mentions his ardent desire to put his computational skills into the service of science and in particular at a German institution, but that he has given up hope that this could become reality and is therefore travelling to England to try and realize his desire there:

Mein innigster Wunsch war es von jeher bei irgend einer Akademie meines deutschen Vaterlandes durch meine Rechenkraft an der Seite eines Mathematikers der Wissenschaft ein bleibendes Denkmal meiner Befähigung hinterlassen zu können; dass derselbe zu verwirklichen wäre, mag vorliegende Arbeit verbürgen. Die Hoffnung, dass mir dies auf deutscher Erde gelingen möchte, musste ich leider aufgeben; ich stehe eben im Begriffe nach England zu reisen, um auf fremden Boden zu versuchen meinen Wunsch zu realisieren.

What exactly transpired next is not clear from the sources I consulted, but that same year (1850) Dase seems to have taken the initiative by consulting with Gauss in person about creating tables of factors.

E. W. Scripture, Arithmetical Prodigies. Worcester: F. S. Blanchard 1891, p. 19:

In 1849 Dase had wished to make tables of factors and prime numbers from the 7th to the 10th million. The Academy of Sciences at Hamburg was ready to grant him support, provided Gauss considered the work useful.

As requested by Dase, Gauss followed up their conversation with a letter dated December 7, 1850 whose contents is reproduced in the preface of the first volume of tables of factors published shortly after Dase's death. In the letter, Gauss summarizes earlier efforts on tables of factors, in particular tables by L. Chernac published in 1811 covering numbers up to 1020000 and tables by J. K. Burckhardt published in 1817 covering the first three million numbers, as well as a manuscript covering the fourth through sixth million by A. L. Crelle (founder of Crelle's Journal), deposited with the Berlin academy which Gauss expects to be published shortly. He therefore recommends that Dase initially tackle numbers from $6000000$ to $10000000$.

It is not clear how exactly the work proceeded after Gauss's recommendation. Securing financing for the table-making effort seems to have been a major issue. In a paper on how best to create factor tables (including estimates of the time and money required) dated May 1853 but published only posthumously in 1856, Crelle himself recommends [p. 92] Dase for the job and mentions that Dase has the ambition to compute factor tables up to 30 million, but raises the issue [p. 93] how Dase could make a living while undertaking this work.

A. L. Crelle, "Wie eine Tafel der untheilbaren Factoren der Zahlen bis zu beliebiger Höhe möglichst leicht und sicher aufzustellen sei." Journal für die reine und angewandte Mathematik, Vol. 51, No. 1, 1856, pp. 61-99 (scan)

W. W. Johnson, "Mr. Glaisher's factor tables and the distribution of primes." Annals of Mathematics, Vol. 1, No. 1, March 1884, pp. 15-23:

In 1860, through the support of the patrons of science in his native town, Hamburg, Dase was enabled to devote himself entirely to carrying out Gauss's project, but in 1861 he died suddenly, leaving the seventh million complete and the eight million almost complete, as well as a great part of the work for the ninth and tenth millions.

A death notice for Dase in a Vienna newspaper also mentions financial support by a committee of Hamburgians ultimately amounting to several thousand marks, enough to allow him to focus on table making for a period of three to four years:

Wiener Zeitung, Sunday, September 15, 1861, p. 4:

Nachdem er nemlich ziemlich plan- und zwecklos umhergestreift war und nur kurze Zeit eine Verwendung durch das preußische Finanzministerium erhalten hatte, kehrte er wieder nach Hamburg, seiner Vaterstadt, zurück, wo sich auf Anregung einiger seiner Gönner ein Komité bildete, welches Geldsammlungen zu dem Zwecke veranstaltete, um Dase 3–4 Jahre hindurch eine sorgenfreie Existenz zu sichern, die er dazu benutzen sollte, um ein logarithmisches Riesenwerk (über die Primzahlen) auszuarbeiten. Anfangs flossen die Beiträge ziemlich spärlich und Dase gab auf den Rath seiner Freunde wiederholt öffentliche Proben seiner Kunst; endlich wurden doch mehrere Tausend Mark zu dem oben gedachten Zwecke zusammengebracht und Dase machte sich an die ihm übertragene Arbeit, die nunmehr leider unvollendet bleibt.

After Dase died unexpectedly from a stroke in 1861 his work was posthumously published in three volumes, with H. Rosenberg completing the tables for the nineth million while the tables for the tenth million were never published:

Zacharias Dase, Factoren-Tafeln für alle Zahlen der siebenten Million, oder genauer von 6000001 bis 7002000, mit den darin vorkommenden Primzahlen. Hamburg: Perthes-Besser & Mauke 1862 (Google scan)

Zacharias Dase, Factoren-Tafeln für alle Zahlen der achten Million, oder genauer von 7002001 bis 8010000, mit den darin vorkommenden Primzahlen. Hamburg: Perthes-Besser & Mauke 1863 (Google scan)

Zacharias Dase and H. Rosenberg, Factoren-Tafeln für alle Zahlen der neunten Million, oder genauer von 8010001 bis 9000000, mit den darin vorkommenden Primzahlen. Hamburg: Perthes-Besser & Mauke 1865 (Google scan)

The following publication offers a useful overview of early efforts in creating factor tables: Denis Roegel, "A reconstruction of the tables of factors of Burckhardt, Dase, and Glaisher (1814-1883), and their extension to the tenth million". Research Report hal-00654420, 2011

So far I have not found any peer-reviewed publication that presents solid evidence that Dase was autistic, however multiple authors speculate in that direction based on what appears to me as rather scant anecdotal details. For example, the article on Dase in MacTutor simply states:

A modern medical expert assessing Dase using the descriptions of his personality and other factors, would suggest that he suffered from Asperger's syndrome.

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  • $\begingroup$ I'm curious why Dase is spelled Dahse in the 1844 publication $\endgroup$ Aug 1, 2023 at 13:42
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    $\begingroup$ @J.W.Tanner In the accompanying letter printed there, Professor von Schulz Strasznicky spells Dahse in the first line, although later he spells Dase several times. $\endgroup$ Aug 1, 2023 at 14:37

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