Thabit ibn qurra and the pair of amicable numbers 17296, 18416

Mathematics and Her Sisters in Medieval Islam: A Selective Review of Work Done from 1985 to 1995

“…The above ''miniature'' of Thā bit at work on a text fits well with Hogendijk's argument [58] that Thā bit not only discovered a rule for generating amicable numbers but used it to find a new pair of amicable numbers (17296 and 18416). Thā bit's theorem on amicable numbers inspired the Persian mathematician Kamā l al-Dīn al-Fā risī (d. ca.…”

A Historical Survey of the Fundamental Theorem of Arithmetic

“…His results appear in Tadhkirat al-Ahbāb fībayān al-tahābb (which means "memorandum for friends explaining the proof of amicability"). His main concern was amicable numbers, and his aim was to prove by a different method the theorem of Ibn Qurra that states "if three numbers p = 3.2 n−1 − 1, q = 3.2 n − 1, and r = 9.2 2n−1 − 1 are prime, and if p, q > 2, then the pair 2 n pq and 2 n r are amicable" [Hogendijk 1985]. Ibn Qurra (836-901) had worked only lightly on the decomposition of integers and combinatorial methods.…”

Perfect, Amicable and Numbers of Equal Weights (Historical Notes)

“…An important consideration here is whether Thabit was familiar with the methods of Archimedes. Al Karkhi 4 also singled out in his book ( Badi fil alhisab ), [9], chapter on the amicable numbers proving Ibn Qurra theorem and concludes the following: If D,E is a pair of amicable numbers,then it is nescerly that one of them is an abundant number and the other is a deficient number, E- [4,6], Ibn Qurra Theorem, and identify the two amicable numbers 17,296 and 18,416 in the case of n = 4,and Ibn Hidor Tadili (d. 1413 AD) identify the same lived and put the most important production in Baghdad at the end of the tenth and the begining of Eleventh century.He has spent apart of his life in the mountainous areas, where he worked in engineering, this work appeare in his book "About drilling of wells,.He "died in Baghdad in (421 AH = 1020 AD), considered by some as one of the greatest Mathematician who have had a real impact in the progress of Mathematical Sciences, he has several books, including: a book in the Indian account, which speaks for the extraction approximate polynomial roots, and a book in the induction , and Alkafi book which contains rules of the product signs and unknowns, sums of the Algebraic terms and the laws of the last term and the total sum in numerical sequence , and the square root of Algebraic amounts. while in his book Alfkhry in algebra he study many problems, he is the first Arab proved that .In his book "Ellal Algeber wa almukabla" and, in het set out rules to solve the equations of the second degree as well as multiplication and division and addition and subtraction rules formulas for two rational numbers and proved those rules algebraically.…”

“…P. Hogendijk (1985) pointed out that Thabit carried through his proof of his theorem for the case when the parameter n = 7 which indicates that Thabit knew the amicable pair 17,296 and 18,416 (the 2 ) Thābit ibn Qurra al-Ḥarrānī( 826 -901) was an Arabic mathematician, physician, astronomer, and translator of the Islamic Golden Age who lived in Baghdad in the second half of the ninth century during the time of Abbasi Caliphate.Ibn Qurra made important discoveries in algebra, geometry, and astronomy. In astronomy, Thabit is considered one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.…”

“…The discovery of the two couples of amicable numbers (17,296 and 18,416); (9,363,584 and 9,437,056) is usually attributed to Fermat and Descartes.But it has been recently shown that Fermat's couple had been calculated by an earlier Moroccan mathematician, Ibn al-Banna 7 , [13,14,20], who calculated it at least a century earlier. This mathematical result was subsequently known to many mathematicians, as was the case for the so called " Descartes couple.…”