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656Edit

656

656 is a positive integer that sits between 655 and 657. In mathematics it is a composite number with a tidy prime factorization and a handful of interesting properties that make it a good teaching example for basic number theory. Beyond its role as a numeral, 656 also appears as a reference point in historical dating, notably as a year in the early medieval period. The way this single number ties together arithmetic structure and the real-world chronology offers a small window into how numbers and history intersect.

Mathematics

  • Factorization and divisors

    • 656 = 2^4 × 41. This makes it a product of a power of two and a distinct prime, a simple yet instructive case for understanding the fundamental theorem of arithmetic.
    • It has 10 divisors: 1, 2, 4, 8, 16, 41, 82, 164, 328, 656. The divisor function d(n) for prime-power factorizations follows the rule (a+1)(b+1)… when n = p^a q^b …, which is easy to illustrate with 656.

  • Totient and sums of divisors

    • Euler’s totient function gives φ(656) = 320, reflecting how many integers up to 656 are coprime to 656.
    • The sum of divisors σ(656) = 1302, which includes 1 and 656 itself; the sum of proper divisors (excluding 656) is 1302 − 656 = 646.

  • Base representations

    • In hexadecimal (base 16), 656 is 0x290.
    • In octal (base 8), 656 is 0o1220.
    • In binary (base 2), 656 is 0b1010010000, a representation that highlights its 2-power structure: 2^4 is present, alongside other powers via the combination 2^9 + 2^7 + 2^4.

  • Context in number theory

    • 656 is an example of a non-prime, composite number with a simple yet nontrivial factorization. It serves to illustrate multiplicativity of many arithmetic functions (like the totient and divisor sums) that are central to introductory number theory.

  • Related concepts

History and calendar relevance

  • The year 656 CE

    • In the year 656 CE, the Muslim world experienced a pivotal event: the assassination of the caliph Uthman ibn Affan. This act precipitated a period of intense political and religious upheaval that culminated in the First Fitna, the first major Islamic civil war, and a reordering of leadership in the early caliphate.
    • Key figures associated with the events around 656 include Uthman ibn Affan, Ali ibn Abi Talib, and Muawiya I. The ensuing power struggles influenced the trajectory of the caliphate and the development of Islamic political theory and practice.
    • The consequences of these events extended beyond a single year, affecting governance structures, succession norms, and the nature of political legitimacy in the Islamic world. The year also aligns with the Hijri calendar as 37 AH, illustrating how different calendrical systems intersect historical memory.

  • Controversies and debates (from a traditional perspective)

    • Historians debate the causes and interpretations of Uthman’s assassination, including questions about governance, nepotism, and rivalries among tribes and factions. Supporters of established rule emphasize the importance of orderly succession, institutional continuity, and the rule of law in maintaining social stability and contract enforcement.
    • Critics, both ancient and modern, have argued that policies perceived as favoring particular groups contributed to discontent and unrest. From a traditional or conservative vantage, it is argued that violence and rebellion undermine the legitimacy and effectiveness of leadership and damage the social fabric that allows markets, property rights, and long-term investment to function.
    • Some modern writers attempt to frame these events through contemporary lenses of identity or grievance politics. From the standpoint reflected here, such interpretations can be seen as anachronistic; they often obscure the primary historical dynamics—political power, governance, and the mechanics of rule—that shaped the era. In that sense, many critics of “presentist” readings argue that an emphasis on moralizing categories risks distorting the historical record and the hard lessons about institutions, authority, and consequences of political upheaval.

  • Calendar and historiography

    • The interplay between the Julian calendar (used in much of 656 CE) and the Hijri calendar reflects how societies frame time differently. The study of this year thus sits at the intersection of arithmetic, chronology, and historiography, illustrating how numbers and dates organize both mathematics and memory.

See also