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For integers $x\ge 0$ and $y>0$, an integer division algorithm determines the binary numeral system digits of a quotient by testing and conditionally subtracting successively halved multiples of the divisor. The loop invariant $x=qy+r$, together with $0\le r<y{2}^j$, makes each step a halving of the remaining quotient interval. Start with $q=0,r=x$ and $j$ large enough that $x<y{2}^j$. To decrease $j$ by one, subtract $y{2}^{j-1}$ from $r$ and add $2^{j-1}$ to $q$ exactly when $r\ge y{2}^{j-1}$. Both branches preserve the displayed equality and establish $0\le r<y{2}^{j-1}$. When $j=0$, the quotient and remainder conditions hold.

The binary numeral system is positional notation with radix two. Each binary digit is zero or one, and its place value is a power of two. A one includes that weight in the sum; a zero skips it. The binary representation $11001101_2$ therefore selects $128,64,8,4,1$ and represents the integer $205$.

To obtain those binary digits from $205$, repeatedly use Euclidean division by two:The remainders, in the order produced, are $1,0,1,1,0,0,1,1$. Read them in reverse order to get $11001101_2$. The final zero is the quotient that stops the conversion, not an additional leading binary digit. Starting with zero itself needs the special single-digit representation $0_2$.

A useful independent check reads the binary digits from left to right. Start with $v=0$ and, for each binary digit $d$, replace $v$ by $2v+d$. For $11001101_2$, the intermediate values are $1,3,6,12,25,51,102,205$. This reconstructs the integer without reversing a list and explains why appending a zero on the right doubles the value.

Binary is well suited to computation because each binary digit has just two possibilities. A stored bit can encode either possibility; groups of bits encode larger integers. Practical integer arithmetic can use a radix that is itself a power of two, grouping several binary digits into each stored numeral digit.^[ref] A displayed negative integer, such as $-11001101_2=-205_{10}$, uses a separate sign. A computer's particular signed storage convention is an additional encoding choice, so the displayed sign should not be confused with an extra binary place.

A fixed-sized, aligned block of consecutive bytes transferred and tracked as a unit by a data memory cache. Here cache line and cache block name that unit of data, including its identity in backing computer memory; a resident slot holds a copy. Reading one element can bring neighbouring elements into the same cache line, exploiting spatial locality.

For a nonnegative integer physical address $a$ and a cache line size $B$ bytes, Euclidean division gives the backing cache block number $q$ and offset $d$:Here the brackets denote the floor function and $\mathrm{mod}$ gives the remainder. Cache line $q$ contains bytes at physical addresses $Bq$ through $Bq+B-1$. With $B=16$, read address 52 selects cache block 3, spanning addresses 48–63, and offset 4. Addresses 48 and 52 therefore request the same cache line; addresses 15 and 16 request different cache lines despite being adjacent.

For a cache line size $B=2^b$, the low $b$ bits give the offset. In a fully associative cache, all remaining bits identify the backing cache block as its cache tag; there are no address bits selecting a particular cache slot. An eight-bit illustration gives $52={00110100}_2$: four bits 0011 for cache block 3 and four bits 0100 for offset 4 in the binary numeral system. The eight-bit width is illustrative; the trace does not assume an actual eight-bit machine. ^[ref]

Distinguish this identity from a cache slot, which can hold different cache blocks over time. After read nine in the memory cache example, cache slot 0 contains cache line 3 and cache slot 1 contains cache line 0. Their cache tags identify 3 and 0; neither cache line number is its slot number. Some texts use “cache line” for the storage entry itself; this article reserves cache slot for that role.

The ceiling function gives the least integer large enough to meet a real lower bound. For a real number $x$,Thus $\lceil 2.7\rceil =3$, $\lceil -2.7\rceil =-2$, and $\lceil 2\rceil =2$. The operation rounds upward in the total order on the real numbers, which can move a negative input towards zero. Its difference from the input obeys $0\le \lceil x\rceil -x<1$. Python's math.ceil implements this least-integer definition. ^[ref].

The ceiling count for fixed-size batches is a direct application. Suppose each box holds at most $k$ items, with positive integer $k$, and there are $N\ge 0$ items. The smallest number of boxes with enough capacity is $\lceil N/k\rceil$: a candidate count $b$ must satisfy $bk\ge N$. For $N=101$, $k=20$, six boxes are needed, leaving $120-101=19$ unused places. Exactly 100 items need only five boxes, and zero items need zero boxes. Adding one to the floor function result unconditionally would therefore overcount at exact multiples of $k$.

For nonnegative integer $N$ and positive integer $k$, Euclidean division gives the exact identityWrite $N=qk+r$, $0\le r<k$. The count is $q$ when the remainder is zero and $q+1$ otherwise, and the right-hand side encodes these two cases. This argument specifies the domain of the identity; arbitrary negative denominators require their own sign conventions.

The ceiling function is a nondecreasing function, constant on each of the half-open intervals $(m-1,m]$, and has a discontinuity of a function at every integer. It is tied to the floor function by $\lceil x\rceil =-\lfloor -x\rfloor$. Adding an integer $j$ shifts its output by the same amount: $\lceil x+j\rceil =\lceil x\rceil +j$. At an integer the floor and ceiling agree; at every other input they differ by one. These statements refer to exact real numbers; an input already perturbed by floating-point arithmetic can land on the other side of a threshold.

The Chinese remainder theorem says that remainders modulo pairwise coprime positive integers can be specified independently. For $m_1,\dots ,m_r$ pairwise coprime and $M=m_1\cdots m_r$, every prescribed list $a_1,\dots ,a_r$ determines one congruence class modulo $M$ satisfying $x\equiv a_j(\mathrm{mod}{m}_j)$ for all $j$. It determines a class, rather than one unrestricted integer: adding a multiple of $M$ preserves every remainder.

For example, combine $x\equiv 2(\mathrm{mod}3)$ and $x\equiv 3(\mathrm{mod}5)$. Write $x=2+3k$ from the first condition. The second becomes $3k\equiv 1(\mathrm{mod}5)$. Since three has modular inverse two modulo five, $k\equiv 2(\mathrm{mod}5)$, giving $x\equiv 8(\mathrm{mod}15)$. The integers $8$, $23$ and $-7$ all work. The fifteen possibilities modulo fifteen correspond to the fifteen pairs in a three-by-five grid.

The theorem also preserves arithmetic: reducing an addition or a multiplication modulo each $m_j$ gives the same tuple as performing that operation on the corresponding congruence classes. In the language of quotient rings, reduction is a ring isomorphismOne large residue is thus an exact encoding of several smaller residues. This is useful for splitting arithmetic into smaller calculations and then reconstructing the result; MIT's modular arithmetic notes develop both the reconstruction and the product representation. ^[ref] For a concrete multiplication, $7$ corresponds to $(1,2)$ and $8$ to $(2,3)$ modulo $(3,5)$. Their componentwise product is $(2,1)$, which reconstructs to $11(\mathrm{mod}15)$, agreeing with $56\equiv 11(\mathrm{mod}15)$.

A constructive formula explains the independence. For $m_j>1$, put $M_j=M/m_j$ and choose $u_j$ with $M_j{u}_j\equiv 1(\mathrm{mod}{m}_j)$, using a modular multiplicative inverse. Then $e_j=M_j{u}_j$ is one modulo $m_j$ and zero modulo every other $m_i$. Thereforehas the desired remainders. A modulus equal to one imposes no restriction and can be omitted. Uniqueness follows because a difference divisible by every $m_j$ is divisible by their product when they are pairwise coprime. This short construction gives a reusable algorithm as well as an existence statement.

The coprime hypothesis explains why every cell in the left grid is attainable. With integer moduli six and eight, a pair must agree modulo their greatest common divisor, two. There are twenty-four compatible pairs, rather than forty-eight arbitrary pairs; each corresponds to a class modulo the least common multiple, twenty-four. The Chinese remainder theorem for noncoprime moduli states the resulting compatibility rule.

For a reduced fraction $p/q$ with $0<p<q$, $q>1$, and $q,10$ coprime integers, its decimal representation repeats immediately after the decimal point. The shortest repeating block has lengthHere $\mathrm{ord}$ denotes multiplicative order, and $\phi$ denotes Euler's totient function. Changing the numerator preserves the decimal period as long as the fraction remains reduced.^[ref]

For example, long division of $1/7$ gives the remainders $1,3,2,6,4,5,1$, with emitted digits $1,4,2,8,5,7$. ConsequentlyEach remainder determines both the next digit and the next remainder; returning to one restarts the same six steps.

More generally, initialize long division with $r_0=p$. At each step, division with remainder givesso $r_j\equiv p10^j(\mathrm{mod}q)$. Since $p,q$ are coprime integers, the modular inverse of $p$ shows that $r_L=r_0$ exactly when $10^L\equiv 1(\mathrm{mod}q)$. To see why the digit block cannot repeat sooner than the remainders, a purely repeating decimal with a block of $L$ digits represents $D/(10^L-1)$ for an integer $D$. A reduced fraction $p/q$ with that value forces $q\mid 10^L-1$. Hence the shortest digit repeat and the multiplicative order coincide.

Knowing the decimal period predicts how many digits must be computed and stored: one block determines the whole repeating decimal. It also constructs the block directly. With $L={\mathrm{ord}}_q(10)$, the integeris the block, padded on the left with zeros to $L$ digits. For $1/11$, $L=2$ and $D=9$, so the block is $09$ and $1/11=0.\overline{09}$. Reduction matters: $7/21=1/3=0.\overline{3}$ has decimal period one, whereas ${\mathrm{ord}}_{21}(10)=6$.

If the reduced denominator contains prime factors two or five, ten has no multiplicative order modulo that whole denominator. WriteThe decimal preperiod has exactly $k_0$ digits. If $m>1$, the remaining repeating decimal has decimal period ${\mathrm{ord}}_m(10)$. For instance, $28=2^2\cdot 7$ gives $1/28=0.03\overline{571428}$: two prefix digits, then six repeating digits. If $m=1$, the decimal representation is a terminating decimal with exactly $k_0$ fractional places. We use the representation ending in zeros, rather than its alternative ending in nines; for example $1/40=0.025$ terminates after three places.

A short derivation explains both cases. After $k$ digits, multiplying the reduced fraction by $10^k$ and discarding its integer part leaves a reduced fraction whose denominator is $q/\gcd (q,10^k)$. This denominator first becomes coprime to ten at $k=k_0$. It then equals $m$, giving an immediate repeat when $m>1$, or an integer when $m=1$. The same long division argument works for positional notation in any base $b\ge 2$: for a reduced denominator $q>1$ coprime to $b$, the shortest repeating block has length ${\mathrm{ord}}_q(b)$.

Count or sum integers by processing their numerical digits and merging prefixes that have the same possible continuations. Typical state records include the current position, whether the prefix still equals an upper limit, a remainder, or a bit mask of digits already used. The final answer comes from accepting exactly those states satisfying the desired digit constraints.

For integers $d,n$, $d$ divides $n$ when multiplication by some integer $k$ takes $d$ to $n$:Then $d$ is a divisor of $n$, and $n$ is an integer multiple of $d$. The notation $d\nmid n$ means that no such integer $k$ exists. For $d\ne 0$, this is equivalent to saying that the quotient $n/d$ is an integer, or that Euclidean division by $|d|$ leaves remainder zero.

For example, $3\mid 12$ because $12=3\cdot 4$, whereas $5\nmid 12$ because $12=5\cdot 2+2$. Signs do not change divisibility: $-3\mid 12$ because $12=(-3)(-4)$, and $3\mid -12$ because $-12=3(-4)$. In general, changing the sign of either input preserves the relation.

The definition here permits zero as either input. Every integer divides zero, since $0=d\cdot 0$. In particular, $0\mid 0$; however, $0\nmid n$ whenever $n\ne 0$, since $0k$ is always zero. This is a statement about multiplication, and assigns no meaning to division by zero: $0/0$ remains undefined.

Two useful properties come directly from the defining multiplication. Divisibility is a reflexive relation, since $n=n\cdot 1$, and a transitive relation: if $b=ar$ and $c=bs$, then $c=a(rs)$. Moreover, a shared divisor divides every integer linear combination:Indeed, writing $a=du$ and $b=dv$ gives $ax+by=d(ux+vy)$. For example, $6\mid 12$ and $6\mid 18$ imply $6\mid (18-12)$. This small observation explains why the Euclidean algorithm preserves common divisors, and why Bézout's identity can certify a greatest common divisor.

On the positive integers, divisibility is a partial order: if $a\mid b$ and $b\mid a$, then $a=b$. The restriction matters, since $3\mid -3$ and $-3\mid 3$ although $3\ne -3$. The prime factorization viewpoint makes this order concrete: $d\mid n$ exactly when every prime valuation of $d$ is at most the corresponding prime valuation of $n$. Thus one relation connects equal grouping, prime factorization, and modular arithmetic.

Book VII, definitions 3 and 5 of Euclid's Elements describe the corresponding positive-number relation through one number measuring another and through integer multiples. The sign and zero conventions above belong to the present integer formulation, rather than to those ancient definitions. ^[ref]

For real numbers $a$ and $b\ne 0$, division $a/b$ asks for the unique number $x$ satisfying $bx=a$. Equivalently, $a/b=a{b}^{-1}$ uses the multiplicative inverse of $b$. This description separates the role of the denominator from the result: the quotient undoes a specified multiplication.

For example, $3/2$ is the number whose double is three, so it is exactly $1.5$. A second interpretation asks how many portions of size $2$ fit into an amount $3$; the answer is one and a half portions. Equal sharing and measuring with a specified portion size lead to the same quotient, although the units assigned to the result can differ.

In the natural numbers, exact division need not be possible. Euclidean division instead keeps a remainder: $17=5\cdot 3+2$, with $0\le 2<5$. Thus seventeen objects form three complete groups of five with two left over. As a rational number, the same quotient is $17/5=3+2/5=3.4$. The remainder and the fractional part describe the same unfinished group in different ways.

Division by zero cannot retain the unique-solution rule. For $a\ne 0$, the equation $0x=a$ has no solution; for $a=0$, every real number $x$ solves $0x=0$, so there is no unique quotient. This explains both failures through multiplication, rather than assigning $a/0$ a value that would contradict ordinary arithmetic. ^[ref]

An equivalence relation says which set elements should count as the same for a particular purpose. In modular arithmetic, for example, distinct integers with the same remainder are equivalent. The relation gathers interchangeable set elements into equivalence classes; the quotient set then treats each whole equivalence class as one object.

Formally, an equivalence relation on a set $X$ is a binary relation $R\subseteq X\times X$, written $x\sim y$, satisfying three conditions for all $x,y,z\in X$:These conditions make the equivalence classes a partition of a set: each set element belongs to exactly one class. ^[ref]

For a worked example, take the finite set $X=\{0,1,\dots ,8\}$ and restrict integer congruence modulo three to $X$:Reflexivity follows because $x-x=0$. The symmetric relation property follows by changing the sign of a difference, and transitivity follows by adding $x-y$ and $y-z$: divisibility by three is preserved in both steps. Thus $0\sim 3$ and $3\sim 6$ force $0\sim 6$. The three equivalence classes are $\{0,3,6\}$, $\{1,4,7\}$ and $\{2,5,8\}$. Their members alternate along the number line; membership in an equivalence class need not mean being close together.

The transitivity requirement rules out a tempting alternative. On $\{0,1,2\}$, define nearness by the absolute value condition $|x-y|\le 1$. This binary relation is reflexive and symmetric, but $0$ is near $1$ and $1$ is near $2$ while $0$ is not near $2$. Its neighborhoods $\{0,1\}$, $\{0,1,2\}$ and $\{1,2\}$ overlap without being equal. They are not equivalence classes or a partition of a set.

For an odd prime $p$ and $p\nmid a$, Euler's criterion identifies the Legendre symbol by modular exponentiation:

Thus a quadratic residue gives one and a quadratic nonresidue gives minus one, represented by $p-1$ if a nonnegative remainder is used. The formula also holds when $p\mid a$, because $(p-1)/2>0$ and both sides are zero modulo $p$.^[ref]

For example, modulo seven the exponent is three: $2^3\equiv 1$ confirms that two is a quadratic residue, while $3^3\equiv 6\equiv -1$ confirms that three is a quadratic nonresidue. Modulo eleven the exponent is five, and $2^5=32\equiv 10\equiv -1$ confirms that two is a quadratic nonresidue. The test answers whether a square root exists; it does not itself return the roots $3,4$ of $x^2\equiv 2(\mathrm{mod}7)$.

A short proof needs only Fermat's little theorem and the root bound for a polynomial. Put $m=(p-1)/2$. If $a=x^2\ne 0$ in the prime field, then $a^m=x^{p-1}=1$. There are already $m$ such quadratic residues. The nonzero polynomial $T^m-1$ has polynomial degree $m$, so it has no other polynomial roots. Finally, for every nonzero $a$, Fermat's little theorem gives $(a^m{)}^2=a^{p-1}=1$. Factoring $(a^m-1)(a^m+1)=0$ in the field shows $a^m\in \{1,-1\}$; every quadratic nonresidue must take the remaining value $-1$. This proves the criterion without first finding a primitive root.

For a larger original example, decide whether seven is a quadratic residue modulo the prime number $101$. Exponentiation by squaring computes the required power without listing fifty squares:Therefore $(7/101)=-1$ and $x^2\equiv 7(\mathrm{mod}101)$ has no solution. Each intermediate product is reduced modulo $101$, as in modular exponentiation. The advantage is that exponentiation by squaring uses a number of multiplications proportional to the binary digit length of the exponent rather than testing every potential square root.

The odd prime hypothesis is essential. For the composite modulus nine, $2^{(9-1)/2}=2^4\equiv 7(\mathrm{mod}9)$, which is neither one nor minus one. Although four is a nonzero square modulo nine, $4^4\equiv 4(\mathrm{mod}9)$, so even the square-to-one implication fails. The Jacobi symbol and Euler pseudoprime discuss different questions for odd composite moduli; they do not make this a universal square test.

For a unit modulo n $a$ and a nonnegative integer $k$, division with remainder writes $k=q\phi (n)+r$ with $0\le r<\phi (n)$. Euler's theorem then givesThis replaces a potentially enormous exponent by a small remainder. The exponent itself is reduced modulo Euler's totient function, while the powers are reduced modulo $n$: these are different integer moduli.

For a worked modulo-fifteen calculation, seven is coprime to fifteen and $\phi (15)=8$. ThereforeUse exponentiation by squaring on the remaining exponent $5=(101{)}_2$, reducing after each multiplication:Thus $7^{2029}\mathrm{mod}15=7$. Modular exponentiation supplies the computation; Euler's theorem supplies the optional shortening of its exponent. Exponentiation by squaring also works for nonunit bases, without using this shortening.

The guaranteed exponent $\phi (n)$ is a common return exponent and an upper bound on the shortest period; it need not be that period. The multiplicative order $d={\mathrm{ord}}_n(a)$ is the least positive return exponent for the chosen unit. The Carmichael function $\lambda (n)$ is the least positive return exponent that works for every unit. Their precise relation isEither smaller return exponent can replace $\phi (n)$ in exponent reduction for units. Modulo fifteen, ${\mathrm{ord}}_{15}(7)=\lambda (15)=4<8=\phi (15)$, so $2029=507\cdot 4+1$ gives $7^{2029}\equiv 7$ immediately. For base four the multiplicative order is only two, smaller even than $\lambda (15)$. A return at exponent eight alone does not prove that the multiplicative order is eight.^[ref]

If the reduced exponent is zero, the rule gives $a^k\equiv a^0=1$, and still requires a unit. Applying it to the nonunit three would falsely predict $3^8\equiv 3^0=1(\mathrm{mod}15)$; the actual residue is six. For negative exponents, first interpret the power using a modular multiplicative inverse, as in modular exponentiation.

A mixed-radix positional notation using factorials as place values: $n={\sum }_{i\ge 1}{a}_{i}i!$ with $0\le a_i\le i$. For lexicographic order on permutations, division by $(k-1)!$ selects the next symbol when $k$ symbols remain; the remainder gives the rank within that block.

The floor function and ceiling function turn a real number into directed integer bounds. They answer two different questions: which integer is the largest one that does not exceed $x$, and which is the smallest one that is not less than $x$?Both outputs are integers. These are exact mathematical operations; they need not choose the closest integer, and they do not refer to roundoff error in floating-point arithmetic.

For example,The fractional part is $\{x\}=x-\lfloor x\rfloor$. The negative example matters: discarding the digits after the decimal point would give $-2$, which is truncation toward zero, rather than the floor function's value $-3$. At an integer, the floor function and ceiling function agree and the fractional part vanishes.

The usefulness is a bridge from continuous quantities to discrete decisions. An integer count in a closed interval uses both bounds. A ceiling count for fixed-size batches counts how many containers or pages are needed; a floor function counts how many complete groups fit. In Euclidean division, the same bounds specify the quotient, while the fractional part records the scaled remainder. A small amount of notation removes many cases that would otherwise have to be stated separately.

The reflection identityfollows by reflecting the ordered integers across zero: the largest one below $-x$ becomes the smallest one above $x$. For every integer $m$, translation gives $\lfloor x+m\rfloor =\lfloor x\rfloor +m$ and $\lceil x+m\rceil =\lceil x\rceil +m$. These formulas turn upper bounds into lower bounds and move examples between neighboring intervals.

For a historical example of the notation in use, Kenneth E. Iverson's 1962 conference paper A Programming Language lists the floor function, ceiling function and residue operation alongside the ordinary arithmetic operations. It states the two defining inequalities above, then extends operations component by component to vectors and matrices. This shows their role as tools for describing algorithms, rather than merely conventions for writing decimal numbers. ^[ref]

The floor $\lfloor x\rfloor$ is the greatest integer not exceeding the real number $x$. Equivalently, $\lfloor x\rfloor =m$ exactly when $m\le x<m+1$, for an integer $m$. The floor function is constant on each such half-open interval and has a discontinuity of a function at each integer.

For nonnegative integers $N$ and positive integers $k$, $\lfloor N/k\rfloor$ is the number of complete groups of size $k$ that fit into $N$ items. With $N=102$ and $k=20$, five complete groups use 100 items. Euclidean division writes $102=20\cdot 5+2$, retaining the two-item remainder rather than treating the quotient $102/20=5.1$ as a number of complete groups.

An integer is a whole number, positive, negative, or zero. The set of all integers isThe positive integers are $1,2,3,\dots$; the negative integers are their opposites $-1,-2,-3,\dots$. Zero is neither positive nor negative. The nonnegative integers include zero and all the positive integers. On a number line, consecutive integers are one unit apart, negative integers lie to the left of zero, and positive integers lie to its right. ^[ref]

The integers let counting describe changes in both directions. If a counter starts at $-3$ and increases by five, it reaches $2$: $-3+5=2$. Decreasing that result by four gives $2+(-4)=-2$. In addition on the number line, the sign of the added integer chooses the direction and its absolute value gives the distance. The additive inverse $-a$ reverses the change made by $a$.

The integers have closure under a binary operation for addition, subtraction and multiplication: whenever $a,b\in \mathbb{Z}$, the results $a+b$, $a-b$ and $ab$ are still integers. Together these operations give the integers their ring structure.^[ref] For instance, $-3+5=2$, $-3-5=-8$, and $(-3)\cdot 5=-15$. General division has no such closure: $1/2$ is a rational number but not an integer. Some divisions are exact, such as $(-6)/3=-2$; division by zero is undefined. Euclidean division instead produces an integer quotient and a remainder, so $7=3\cdot 2+1$ retains the information that ordinary division puts into the fraction $7/3$.

An integer is the value, whereas its written numeral is an encoding. The decimal representation $205_{10}$ and the binary representation $11001101_2$ name the same integer. Changing positional notation changes how that value is displayed and how calculations are organized; it does not change the value itself.

Integer arithmetic computes with integers. Addition, subtraction and multiplication preserve integer values; division can instead return a quotient and remainder through Euclidean division. For example, $102=20\cdot 5+2$ gives five complete groups and a remainder of two. The ceiling count for fixed-size batches requires six groups to hold all 102 items.

Mathematical integers have no fixed size bound. A computer representation may have one, so an exact arithmetic formula still requires sufficient space for its inputs and intermediate values. The floor function and ceiling function define mathematical bounds independently of a chosen representation.

Division of integers that returns an integer quotient rather than an exact fraction. A rounding convention is required for negative operands. For $n\ge 0$ and $d>0$, the quotient is $q=\lfloor n/d\rfloor$ and the remainder is $r=n-qd$ with $0\le r<d$.

Every nonnegative integer is a sum of squares with four terms, with zero terms allowed. Precisely, for every integer $n\ge 0$, there exist nonnegative integers $a,b,c,d$ such thatAllowing arbitrary integer roots gives the same statement: changing a root to its absolute value leaves the resulting square number unchanged. Four is a uniform upper bound on the number of nonzero terms, rather than a demand for four positive terms. Repeated square numbers are allowed. Conrad states the positive case and explicitly explains the role of zero terms; $n=0$ follows by taking all four roots zero. ^[ref]

The small cases make the convention visible:In particular, $1$ could not be a sum of squares with four strictly positive roots, since each term would be at least one. Geometrically, a square of integer side length $a$ has area $a^2$: the theorem says that any nonnegative whole-number total area can be split into at most four such square areas. It does not assert that those squares tile one larger square or a prescribed rectangle.

Why does the fourth term matter? Two square numbers do not suffice for $3$: in modular arithmetic, a square number has remainder zero or one modulo four, so their sum cannot be three modulo four. The sum of two squares theorem gives a complete criterion for the positive integers representable with two terms. Even three terms leave gaps. A square number has remainder $0,1,$ or $4$ modulo eight; adding three of these, with repetition allowed, never gives seven modulo eight. Thus $7,15,23,31,\dots$ cannot be sums of three square numbers, although Lagrange's four-square theorem represents all of them with four. This also proves that the bound four cannot be reduced to three.

The modulo-eight obstruction is not the only way that three terms can fail. For example, $28$ needs four: if three square numbers added to $28$, their remainders modulo four would force all three roots even. Dividing by four would then express $7$ as a sum of three square numbers, contradicting the preceding calculation. These short checks explain the examples without requiring a full classification of sums of three square numbers.

The practical benefit of Lagrange's four-square theorem is an existence guarantee for a family of Diophantine equations: for every nonnegative integer $n$, the displayed equation has an integer solution. It does not specify a unique solution. For instance, $65=8^2+1^2+0^2+0^2=7^2+4^2+0^2+0^2=6^2+4^2+3^2+2^2$. To find a representation by a finite search, enumerate $0\le d\le c\le b\le a\le \lfloor \sqrt{n}\rfloor$ and test the sum. The floor function supplies a finite bound because no single term can exceed $n$, and sorting roots removes duplicate orderings without discarding any possible sum. The theorem guarantees that the search succeeds.

A greedy algorithm that repeatedly subtracts the largest available square number need not find four terms: for $23$ it chooses $16+4+1+1+1$, while $23=9+9+4+1$ uses four. The appeal of the theorem is this gap between local choices and a global promise: individual square numbers become farther apart, and two or three terms leave arithmetic obstructions, yet four terms cover every nonnegative integer.

For historical orientation, Conrad dates Lagrange's four-square theorem to $1770$ and identifies the memoir as Démonstration d'un théorème d'arithmétique: it appeared in the Berlin Academy's volume for the year $1770$, published in $1772$, pages $123$–$133$. Keeping the volume year and publication year distinct avoids an apparent conflict between dates. ^[ref]

A least-significant-digit radix sort processes numerical digits from smallest to largest place value. Fix integers $R\ge 2$ and $d\ge 1$, and take $n$ complete sorting records whose sort keys satisfy $0\le k<R^d$. Use exactly $d$ positions in positional notation with radix $R$, including leading zeros. At pass $j=0,\dots ,d-1$, apply a stable digit sort to the current array usingThe floor function selects the integer quotient, and the remainder selects the current numerical digit. Move the complete sorting record, including its associated data. There are exactly $d$ passes, so this procedure terminates even for an empty array. The radix sort suffix invariant supplies its algorithm correctness argument.

For an original input, let the letters identify the attached data:Here $R=10$, $d=3$, and leading zeros only align place values; $005$ represents the integer five. The exact pass ledger is:After the units pass, the numerical digits are $0,2,4,5,5,7,7,7$. The ties retain $C$ before $F$ and $B$ before $E$ before $G$. This array is not yet ordered by its whole integer sort keys: $314$ still precedes $305$. After the tens pass, the two-position suffixes are $05,05,14,17,27,27,40,42$. In particular, $217_E$ now precedes $027_B$: their tens numerical digits differ, so stable sorting does not require retaining their previous order. The hundreds pass produces the final sorted array, with all attached letters preserved.