Ibn al-Haytham's factorial remainder construction

MacTutor presents Ibn al-Haytham considering a number that leaves remainder1 on division by 2,3,4,5 and 6, but is divisible by 7. It explains a construction multiplying the consecutive integers and adding one:6!+1=721. This is a particular number-theoretic contribution, not the invention of every remainder problem or every set of simultaneous conditions ¹.

The account invokes what is now called Wilson's theorem and reports a second method giving all solutions, without reproducing its details. We distinguish that report from the museum's independent modern family 301+420j. Nor is 721 the least positive answer:301 is smaller and meets exactly the same conditions ^{1 2}.

A remainder is what remains after removing a whole number of multiples of the divisor. Saying that 721 leaves1 on division by 6 means 721=6×120+1, not that its quotient is 1. The problem requires several conditions at once. Meeting one condition alone is not a fully checked solution ¹.

Factorial is our modern shorthand for multiplying successive positive integers:6!=1×2×3×4×5×6=720. The exclamation mark is mathematical notation, not punctuation copied from Ibn al-Haytham's original. We explain the relationship in language accessible today while keeping the reported historical construction separate from our way of checking it ³.

MacTutor calls the problem a special case of the Chinese remainder theorem. It would therefore be wrong to credit Ibn al-Haytham with every idea of combining division conditions. This record identifies a specific factorial construction within a wider problem tradition. Earlier foundations do not erase a later contribution, but they prevent a story of one person starting from nothing ¹.

This question also differs from his fourth-power volume method and circular reflection-point problem. We do not count one contribution repeatedly under different names. Here the constraints concern integers, not an optical surface or a volume. Working across subjects does not itself establish a direct transfer of a particular method between them ¹.

MacTutor refers to Opuscula and uses a translation in Rashed's study before describing two methods. It explains the first construction and says the second supplies all solutions of this type. We read that account, not the Arabic original or Rashed's full edition. We invent neither an Arabic treatise title nor an ancient book identity from a modern collection label ⁴.

The passage gives no securely dated teaching session or writing room. The author's late tenth- and early eleventh-century lifetime remains a broad horizon; Cairo is biographical context, not a confirmed composition site for this question. A date associated with the Book of Optics is not automatically a date for every other numerical idea ^{5 6}.

The author's existing individual identity basis, drawn from the Optics opening and TDV, is retained alongside the separately declared Sunni frame. We infer no new denomination from his Egyptian employment or a successful calculation. This session is not a fresh critical investigation of every biographical text ^{7 6}.

Wilson is the modern name attached to the arithmetic result. MathWorld discusses Wilson, Waring, Leibniz's earlier knowledge and Lagrange's proof; MacTutor describes an earlier use by Ibn al-Haytham. Both historical accounts remain visible without inventing a direct manuscript-to-manuscript route or assigning the whole history to one person ^{8 1}.

Every integer from 2 through6 is a factor of 720. Division of 720 by any of them leaves0, so division of 721 leaves1. That alone does not establish divisibility by 7. The result now called Wilson's theorem supplies the prime condition: for primep, the number(p−1)!+1 is divisible byp. Here721=7×103 ^{2 1}.

This does not make 721 prime: its factors7 and 103 show the opposite. The prime condition concernsp, not the constructed answer. Nor can7 be replaced silently by an arbitrary composite: for 4, the construction3!+1 gives 7, which is not divisible by 4. A formula valid under a hypothesis cannot be used after that hypothesis is discarded ⁹.

Here is an independent modern derivation of all positive answers, not a recovery of the historical second method. The least common multiple of 2 through6 is 60, soN=60k+1. Since 60 leaves remainder4 on division by 7, we need4k+1 divisible by 7. This requiresk=5+7j, givingN=301+420j.

Takej as a nonnegative integer for positive answers. The first are 301,721 and 1141. For example,301=5×60+1=7×43, while 721=12×60+1=7×103. Listing a few answers does not alone prove completeness. Here completeness comes from reducing the conditions toN=60k+1 and solving the remaining condition modulo7, rather than stopping a search at an attractive number.

The increment 420 explains repetition: it is divisible by 60 and 7, preserving every required remainder. Adding only 7 to 721 gives 728, still divisible by 7 but leaving2 rather than 1 on division by 6. Adding60 preserves the earlier conditions, not automatically the condition for 7. An arbitrary increment therefore does not produce another answer. This checks the common period in our modern solution, not a list found in a manuscript or proof that our formulation reproduces the historical second method. Preserving one constraint differs from preserving the whole system.

The sources read here provide no manuscript route directly connecting the author with Wilson. A modern name and an older result do not establish copying, translation or citation. Reports remain attributed to their sources, and an undocumented transmission link does not become a confident arrow in a diagram ^{1 8}.

Reporting a second method also does not establish its adoption in every school or market. A possible application is not a historical circulation statistic. Modern explanations are evidence of present-day teaching; connecting them to a particular old curriculum would require evidence absent from the passages consulted here.

The immediate ingredients are multiplying the integers and adding one, together with a particular condition for the final divisor. Without that final condition, answers of the form60k+1 remain, rather than a unique number. Repeated factors make 720 a suitable common multiple, but not the least common multiple. Multiplying everything is not always the shortest construction ¹.

We use60 in the modern explanation because it combines the conditions for divisors2 through6. Those divisors are not pairwise coprime:2,4 and 6 have common factors. We must not silently apply a formula restricted to pairwise-coprime moduli. Identifying the actual structure matters more than invoking the broad theorem's name.

MathWorld states the modern theorem as a necessary and sufficient primality condition forp>1. That modern explanation is not an announcement that we found both directions fully proved in Ibn al-Haytham's original. Using a result, proving it and designing a fast practical test from it are separate claims ^{3 1}.

For teaching, one can multiply step by step while retaining only the current remainder, avoiding enormous written integers. This is a modern way to check examples, not a computer algorithm assigned to the medieval author. Similar arithmetic does not establish a direct lineage to every present software or cryptographic application.

The consulted passages record no merchants' time savings or contracts applying this example. The supported contribution belongs to research on simultaneous numerical conditions and multiple solutions. We invent no commercial anecdote to make the arithmetic an ancient everyday device; a question does not need a measured financial return to deserve explanation ¹.

Today's reader gains a clear lesson: a correct answer need not be the smallest, and one answer is not every answer. A constructive method can produce a large, easily checked number while reformulating the conditions gives a shorter description of the entire family. That is a lesson from 301 and 721, not evidence of the author's personal economic motive.

The problem makes the author's non-optical mathematics visible: integers under simultaneous constraints, not only mirrors and eyes. Yet the fame of the Optics does not prove priority in every other subject. Attribution here remains bounded to a particular construction described in the historical account ¹.

It also separates construction, verification and completeness. One can construct 721, check each condition, then ask for smaller answers and the whole family. Those are different tasks. Reporting a second method shows that the historical account does not stop with producing the first number. We do not fill its missing details with a modern proof and conceal the substitution ⁴.

Declared Sunni frame: Ibn al-Haytham's existing individual identity basis is retained, not deduced from arithmetic. Academic account: MacTutor describes the earlier construction and use; MathWorld describes the theorem's name and later European history. Neither establishes our own direct transmission route or makes a modern name the limit of the idea's history ^{6 1 8}.

“Improvement” protects the attribution's boundary: not the first Chinese remainder theorem, first factorial or every congruence calculation. We claim neither original-text reading nor a final ruling on every priority question. What we can show is a particular construction, its limits and a modern statement that helps readers check it.

The image has two grey clocks labelled0,3,6,9, with+4h between them. The hand moves from 9 to 1:9+4=13, which leaves1 modulo 12. This is a modern analogy for cyclic arithmetic, not Ibn al-Haytham's clock or a manuscript illustration. Nor is it the prime7 problem itself:12 is composite ¹⁰.

Congruence does not mean 13=1 as ordinary integers. It means that they leave the same remainder on division by 12. Remainder arithmetic is not erasing the difference between numbers. The picture helps explain the vocabulary, not prove every prime-number property needed by the factorial construction ¹¹.

We read the specified MacTutor passages and MathWorld's definition, first formula and historical paragraph. We did not read the original Opuscula or Rashed's full edition, and do not infer an Arabic manuscript ID from that label. Not creating a new work record here preserves that boundary; it does not assert that the author wrote nothing about the problem ⁴.

The actual Commons image and rights were inspected. Spindled made the original upload on 19October2006; the current version includes a2008 text-alignment correction and carries CC BY-SA3.0. The current original SVG is published unmodified. Upload dates do not date the ancient mathematics or prove a surviving historical clock of this appearance ¹².

An invention story may favour a single name, triumphant scene and final number. This question becomes richer when a particular answer remains distinct from a general description, and a theorem's name from its earlier uses. That is not a judgement against named mathematicians; it is how the museum displays what the evidence does and does not establish ^{1 8}.

The explanation needs neither an imaginary portrait nor a fictional teaching gathering. Read the conditions, check them again, and recognize that the two-clock image belongs to another period. An explicitly declared comparison teaches without making a modern educational asset pretend to document an ancient scene.

Write721 and divide by 2,3,4,5,6, then7. Record remainders, not only quotients. Repeat with 301. The smaller number also meets the conditions. Do not stop at the first correct answer or call it smallest without a search or proof.

Now try7, obtained from 3!+1: is it divisible by 4? No. Ask which hypothesis differs from the original prime7 example. The modern clock was not necessarily in the author's hands; it is only a remainder analogy. Historical evidence and today's arithmetic meet without becoming the same thing ^{2 1}.

• MacTutor: reported remainder problem and construction ¹.
• MathWorld: formula, history and primality condition ¹³.
• TDV: author's setting, not a verified composition site ⁶.
• Commons: modern image and rights ¹².