Thābit ibn Qurra

Thābit ibn Qurra was born in 836 in Ḥarrān, in Mesopotamia (modern Turkey), and died on 18 February 901 in Baghdad. MacTutor records these dates precisely for his death and gives 836 for his birth; Britannica gives "c. 836," and some sources record an alternative "826 or 836," with 836 as the more widely accepted figure. He belonged to the Sabian community, was a native Syriac speaker, and became court astronomer in Baghdad under the Abbasid caliph al-Muʿtaḍid.

Principal treatises

• Book on the Determination of Amicable Numbers, defines perfect and amicable numbers and proves the general theorem discussed below.
• Book on the Composition of Ratios, extends arithmetic operations to ratios of geometric magnitudes. MacTutor connects this to the later generalisation of the number concept.
• Kitāb fī'l-qaraṣṭūn, a work on statics that proves the equilibrium principle of the lever.
• An unnamed geometric work generalising the Pythagorean theorem to arbitrary triangles (MacTutor notes that Pappus independently produced a similar generalisation).
• A work on parabolas and paraboloids that computes, in effect, the integral of \(\sqrt{x}\) using upper and lower sums, and gives an exhaustion method proof that for the first time divides the integration interval into unequal parts, described by MacTutor as a genuine step toward integral calculus.

The amicable number theorem

Two positive integers \(a\) and \(b\) form an amicable pair if the sum of the proper divisors of \(a\) equals \(b\) and vice versa. Thābit's theorem, quoted here in MacTutor's wording:

for \(n > 1\), let \(p_n = 3\cdot 2^n - 1\) and \(q_n = 9\cdot 2^{2n-1} - 1\). If \(p_{n-1}\), \(p_n\), and \(q_n\) are all prime, then \(a = 2^n p_{n-1} p_n\) and \(b = 2^n q_n\) are amicable numbers.

This is pure number theory built in exactly the style of a modern olympiad number theory problem: identify a structure, state a general parametrisation, and prove it holds under stated conditions. MacTutor reports Jan Hogendijk's finding that "Thābit was probably the first to discover the pair of amicable numbers 17296, 18416", attributed to a named scholar and hedged with "probably," which is how we cite it here.

The generalised Pythagorean theorem

Thābit's geometric generalisation extends the Pythagorean identity from right triangles to arbitrary triangles. The relationship it produces is the standard geometric ancestor of the law of cosines construction used in olympiad triangle problems with general angles.

A note on Pythagoras

Thābit's own treatise credits the study of perfect and amicable numbers to Pythagoras. MacTutor flags this attribution explicitly as "probably first made by Iamblichus... and almost certainly false." We preserve that qualification because it illustrates how medieval mathematical texts were sometimes generous, in error, about their own predecessors.

Related curriculum links in the curriculum

Thābit's work is the point of contact between MOTA's number theory strand (via the amicable numbers theorem) and its geometry strand (via the generalised Pythagorean theorem). He is also part of the Heritage overview on the About page.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Thābit ibn Qurra biography (birth 836 in Ḥarrān, death 18 February 901 in Baghdad, patron al-Muʿtaḍid, exact statement of the amicable number theorem, Hogendijk attribution for the pair 17296/18416, Pappus parallel, Pythagoras attribution flagged as false). https://mathshistory.st-andrews.ac.uk/Biographies/Thabit/
2. Encyclopædia Britannica, Thabit ibn Qurrah biography (secondary verification of c. 836 birth date and Syria origin; independent listing of his geometric and number theory writings). https://www.britannica.com/biography/Thabit-ibn-Qurrah