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A mathematician whose work includes number theory and elliptic integrals.

Biography: mathshistory.st-andrews.ac.uk/Biographies/Legendre/

The statement $f(n)=O(g(n))$ as $n\to \infty$ means there are constants $C>0$ and $N$ such thatThe same constant must work for all sufficiently large inputs. Near a finite limit point $a$, replace $n\ge N$ by $0<|x-a|<\delta$ within the domain. Absolute values allow signed or complex-valued functions; when discussing nonnegative operation counts they can be omitted.

The notation records an upper bound, not an exact leading coefficient or necessarily a tight growth order. For example, $3n^2+7n+4=O(n^2)$ because for $n\ge 1$ it is at most $14n^2$. It is also $O(n^3)$, a weaker statement. A tight comparison uses big theta notation; a ratio tending to one uses asymptotic equivalence. Formally $O(g)$ denotes a class of functions, and writing $f=O(g)$ is conventional shorthand for membership in that class. ^[ref]

The first panel compares stipulated operation counts $A(n)=1000n$ and $B(n)=n^2$. Although $A$ is linear and $B$ quadratic, $B(n)<A(n)$ for $1\le n<1000$; the ordering reverses beyond their crossing. These are mathematical cost models, not measured runtimes. The time complexity notation describes the growth of whichever cost function has been specified, whether that function represents a worst-case bound, an average, or another explicit quantity.

The middle panel gives a concrete eventual bound: $n^2+10^{5}n\le 2n^2$ for every $n\ge 10^5$. Its normalized ratio approaches one, but the threshold matters. The final panel shows the asymptotic comparison of exponential functions, with its geometrically shrinking relative difference.

Constants may depend on fixed parameters. For each fixed $\alpha >0$, $f_{\alpha }(n)=\alpha n$ is $O(n)$ with $C=\alpha$; one constant cannot cover every positive $\alpha$. A uniform big O bound makes this extra quantifier explicit.

The notation was already used in nineteenth-century number theory. In his 1976 paper, Donald E. Knuth reports Landau identifying Bachmann's 1894 book as the first appearance of $O$ known to him. Knuth's account gives the earlier mathematical setting of a notation now also common in time complexity analysis. ^[ref]

A mathematician known for work in number theory.

Biography: mathshistory.st-andrews.ac.uk/Biographies/Kummer/

A Fermat number is the positive integer $F_n=2^{2^n}+1$, indexed by a nonnegative integer $n$. The first five are $3,5,17,257,65537$, all prime numbers; the next is the composite number $F_5=4294967297=641\cdot 6700417$. This family offers a compact lesson in number theory: a very persuasive finite pattern can fail, while an exact divisibility relation settles a gigantic case with little computation.

Why does the exponent itself have the form $2^n$? Suppose $m=ab$ is a positive integer with an odd integer factor $b>1$. The polynomial factorization $x^b+1=(x+1)(x^{b-1}-x^{b-2}+\cdots -x+1)$ shows that $2^a+1$ is a proper divisor of $2^m+1$. Consequently, if $2^m+1$ is a prime number, $m$ must be a power of two. That is a necessary condition, rather than a promise that the surviving candidates are prime.

The Fermat number product identity makes distinct Fermat numbers coprime integers. Thus every term has a prime factor absent from all earlier terms, whether or not that term is itself a prime number. This gives an explicit route to infinitely many different prime numbers, without establishing that infinitely many terms of this particular sequence are prime. Meanwhile the prime divisor of a Fermat number criterion filters possible factors through a small multiplicative order calculation.

The family also links arithmetic to Euclidean geometry: a constructible regular polygon can have a number of sides equal to a power of two times a product of distinct Fermat primes. The Gauss-Wantzel theorem gives the precise criterion. For instance, $15=3\cdot 5$ and $17$ qualify, but $9=3^2$ does not. The word "distinct" matters. Carl Pomerance describes this connection in Primality testing: then and now by Carl Pomerance (slide 18).

Historically, Leonhard Euler records Pierre de Fermat's proposed primality pattern, then refutes it by finding the factor $641$ of $F_5$. His paper was presented in 1732 and published in 1738; those are different events, not competing dates for one publication.^[ref]

A short modern check needs only two representations of $641$:In modular arithmetic these give $5^4{2}^{28}\equiv 1$ and $5^4\equiv -2^4(\mathrm{mod}641)$. Substitution yields $2^{32}\equiv -1(\mathrm{mod}641)$, so $641\mid F_5$. This verifies the factor directly; it does not claim to reconstruct Euler's search process.

An integral domain is a nonzero commutative ring without zero divisors. Equivalently, a product of two nonzero elements is always nonzero. The zero ring is excluded, and $\mathbb{Z}/6\mathbb{Z}$ fails because $2\cdot 3=0$. Every field is an integral domain, but the integers show that the converse fails: $2$ has no multiplicative inverse in $\mathbb{Z}$. See MIT's Basic Properties of Rings, definitions 15.9–15.10 and example 15.11.

The practical benefit is multiplicative cancellation. In the integers, $2x=6$ has at most one solution and indeed $x=3$ works. Yet $2x=1$ has no integer solution. Thus multiplicative cancellation guarantees uniqueness when a solution exists; it does not guarantee division by every nonzero element. Passing to the fraction field $\mathbb{Q}$ supplies the missing solution $1/2$.

The polynomial ring $\mathbb{Z}[X]$ is also an integral domain. For example, the product of $2X+1$ and $3X-4$ has leading coefficient $6$ and cannot vanish. Generally the leading coefficient of a product is the product of the two leading coefficients; an integral domain prevents that coefficient from vanishing. This connects integer arithmetic to polynomial arithmetic through one reusable principle.

Historically, this connection grew far beyond elementary examples. Keith Conrad identifies the ring of integers of a number field and a ring of polynomial functions on a smooth plane curve as early important examples of Dedekind domains. This shared structure connects number theory with algebraic geometry; see Conrad's Ideal Factorization, section 6.

A real number is normal in an integer base $K\ge 2$ when every finite word of digits occurs in its positional notation with the same limiting digit block frequency as every other word of the same length. Write its fractional part as $x=0.d_1{d}_2\dots$ in base $K$, using the expansion that is not eventually all $K-1$ digits when two expansions are possible. For every length $r\ge 1$ and every word $w\in \{0,\dots ,K-1{\}}^r$, the requirement isOccurrences overlap: the word $11$ occurs twice in $111$. There are $K^r$ possible words of length $r$, so $K^{-r}$ is the equal share for each. In a decimal representation, this means frequency $1/10$ for each digit, $1/100$ for each pair, and $1/1000$ for each triple, continuing through every finite length. The definition agrees with that used by Copeland and Erdős in their work on explicit normal numbers. ^[ref]

The appeal is that a precise, deterministic real number can have the same limiting digit block frequencies as repeated draws from a uniform distribution on a finite set, with independence between draws. Champernowne constant is an explicit example. Normality is useful for studying digit distribution, but balanced digits alone establish only a simply normal number, and a finite sample cannot establish any of these infinite limits.

Émile Borel introduced normal numbers and established their prevalence in his 1909 paper applying probability to number theory; the stronger all-base statement is discussed under absolutely normal number. ^[ref]

The passage belongs to the English reception of Euclid's number theory. Henry Billingsley's The Elements of Geometrie by Henry Billingsley, published in 1570, brought the Elements into English with extensive explanatory material. The original title page advertises translation and annotations; the extract's worked examples show what explanatory prose could add to a compact definition. Henry Billingsley's numbering calls this Definition 23 of Book VII; the familiar modern numbering is VII, Definition 22. The difference is editorial, not a difference in the perfect number being defined.^[ref][ref]

Here a number's parts are its positive proper divisors, rather than arbitrary pieces of a sum or all smaller integers. In modern notation the definition iswhere $\sigma$ is the sum-of-divisors function. Thus $12$ is not a perfect number: its positive proper divisors total $1+2+3+4+6=16$. It is essential to exclude $n$ itself: including it would double the required sum. The pairing of a divisor $d$ with the fraction $1/(n/d)$ of the whole explains the extract's language of halves, thirds and so on. Its examples are complete checks because they identify every positive proper divisor, not merely some convenient decomposition.

The larger mathematical context is the construction in Book IX, Proposition 36. If $q=2^p-1$ is a prime number, then $N=2^{p-1}q$ is a perfect number. To reconstruct the argument, prime factorization gives all its divisors as $2^j$ and $q{2}^j$, with $0\le j\le p-1$. A finite geometric series therefore givesThe choices $p=2,3,5,7$ produce $6,28,496,8128$. The primality assumption matters: choosing $p=4$ produces $120$, whose sum-of-divisors function is $360$, not $240$. The construction gives a reason for the examples and connects perfect numbers with Mersenne primes.^[ref]

The extract itself supplies a definition and examples, not a classification or a mathematical proof that such numbers are rare. In particular, the converse that every even perfect number has this form belongs to the later Euclid-Euler theorem; it should not be read back into this passage. The historically significant feature is the transmission of deductive arithmetic through vernacular explanation: a book called geometry also teaches the divisibility theory needed for a general construction.

Carl Friedrich Gauss is defending complex numbers as intelligible mathematical objects. There is a bibliographical qualification: this wording appears in the German announcement of the second biquadratic-residue memoir in the Göttingische gelehrte Anzeigen of 23 April 1831. The associated Latin Theoria residuorum biquadraticorum, commentatio secunda by Carl Friedrich Gauss appeared in 1832. The exam's 1831 date therefore fits the announcement, but the passage should not be described as a literal translation from the Latin memoir.^[ref][ref]

The proposed directional vocabulary is mathematically specific. Represent $z=a+bi$ by the point $(a,b)$ in the complex plane. The addition of complex numbers adds components. Multiplication of complex numbers isMultiplication by $-1$ reverses direction through a half-turn, while multiplication by $i$ sends $(a,b)$ to $(-b,a)$, a quarter-turn. Consequently $i^2=-1$ says that two quarter-turns make a half-turn. More generally the moduli multiply and the complex arguments add. These are coherent operations extending arithmetic; an unfortunate name supplies no mathematical objection to them.

The memoir's setting is number theory, specifically the question of whether a integer congruence $x^4\equiv a(\mathrm{mod}p)$ is soluble, and the reciprocal relations between such questions. Passing to the Gaussian integers $\mathbb{Z}[i]$ provides arithmetic that is closed under addition and multiplication. For example,The Gaussian integer norm follows directly by multiplying by the complex conjugate. Here it gives $N(2+i)=5$ and $N(5)=25$, consistent with the factorization. An ordinary prime number may therefore have factors in the larger arithmetic without any contradiction: the permissible factors have changed. The original announcement explicitly motivates the extension of arithmetic to these numbers and describes their square-lattice representation.^[ref]

It is helpful to separate two domains. Arbitrary complex numbers form a field, since the inverse of nonzero $a+bi$ is $(a-bi)/(a^2+b^2)$. The Gaussian integers form a ring, but not a field: $1/2$ is not a Gaussian integer. Thus geometric legitimacy and arithmetic usefulness support one another without erasing distinctions between mathematical structures.

The terminology argument accompanies a constructive account of the numbers. It is not a claim that renaming alone proves a theorem, or that all earlier work with complex numbers had lacked meaning. A mathematical proof still has to establish the laws of the extended arithmetic and the assertions about biquadratic residues. Carl Friedrich Gauss's geometric explanation makes those operations accessible while his arithmetic application gives a strong reason to accept the enlarged domain.