A positive integer $n$ for which some division $n=dq+r$ with integers $d,q,r>0$ and $r<d$ has $d,q,r$ as consecutive terms of a geometric progression in some order. Writing a reduced common ratio as $a/b$ with $a>b>0$ and $\gcd (a,b)=1$, the three terms have the form $t{b}^2,tab,t{a}^2$ for a positive integer $t$.

A reversible number is a positive integer whose sum with its digit reversal has numerical digits that are all odd integers. Here the radix is ten, and reversing must preserve the digit length: the original number cannot end in zero, because that would create a leading zero after reversal.

For example, $36+63=99$, so both $36$ and $63$ satisfy the rule. The example $409+904=1313$ shows that internal zeros are allowed. In contrast, $10$ is excluded by the reversal-length condition, even though $10+1=11$ has odd digits.

The arithmetic carries couple mirrored digit positions. In $409+904$, the units sum $9+4=13$ creates a carry, which makes the next digit $0+0+1=1$. Treating the digit pairs independently would miss this dependence. A counting method can instead track carries from both ends and match them in the middle, avoiding enumeration of every integer of a given length.

A sqube is a positive integer $p^2{q}^3$ for distinct prime numbers $p,q$. The ordering matters: $3^2\cdot 2^3=72$, whereas $2^2\cdot 3^3=108$. By unique prime factorization, the exponents identify the ordered pair $p,q$ from the resulting integer.

Every sqube is an Achilles number. It is a powerful number, because both prime exponents are at least two. It is not a perfect power, because the greatest common divisor of the exponents $2,3$ is one. This connects the square-cube construction to two general factorization classes.

An Alexandrian integer is a positive integer $A=pqr$ for nonzero integers $p,q,r$ satisfyingThe factors may be negative. For example, $A=6$ arises from $(p,q,r)=(1,-2,-3)$, since $1-1/2-1/3=1/6$.

A useful parametrization isIndeed, the signed triple $(n,-n-d,-n-e)$ gives $pq+pr+qr=de-n^2=1$.

Conversely, a valid triple cannot be entirely positive, because then $pq+pr+qr\ge 3$. Since its product is positive, exactly one factor is positive. Write it as $(n,-x,-y)$, so $(x-n)(y-n)=n^2+1$. Both differences have the same sign; they cannot both be negative, because their product would be at most $(n-1{)}^2$. Thus $d=x-n$ and $e=y-n$ are positive and give the parametrization. Enumerating factor pairs of $n^2+1$ therefore produces the candidates without searching arbitrary signed triples.

A nonnegative integer whose positional notation reads the same forwards and backwards, with no leading zeros. This property depends on the base: $585$ is a palindromic number in both decimal representation and binary representation, since $585=(1001001001{)}_2$.

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.

In decimal representation, a bouncy number is a positive integer whose sequence of numerical digits is neither nondecreasing nor nonincreasing from left to right. Equivalently, one adjacent pair strictly rises and another strictly falls. For example, $132$ is bouncy, while $124$ and $431$ are not. Repeated numerical digits are permitted by the weak order conditions: $112$ is nondecreasing.

A number satisfying both weak order conditions has every numerical digit equal, so the overlap consists exactly of the positive repdigits, such as $444$. If $I(N),D(N),R(N)$ count the nondecreasing, nonincreasing and repdigit positive integers at most $N$, respectively, inclusion-exclusion gives the count of bouncy numbers asThis reduces counting the complement of two monotonicity conditions to counting those two simpler families and their explicit overlap.

Golden ratio base is a positional notation with base $\phi =(1+\sqrt{5})/2$. A finite digit string represents ${\sum }_k{d}_k{\phi }^k$, usually using digits $0,1$ and allowing both positive and negative powers.

The identity ${\phi }^2=\phi +1$ gives the rewrite $100_{\phi }=011_{\phi }$, so representations need a normalization convention. In particular, $2=\phi +{\phi }^{-2}=10.01_{\phi }$. Avoiding adjacent ones removes many finite ambiguities. This links an irrational numerical base to Fibonacci numbers and to symbolic rules for rewriting digit strings.

Scientific notation writes a nonzero real quantity as $a\times 10^n$ with an integer $n$ and $1\le |a|<10$. For example, $0.00072=7.2\times 10^{-4}$ and $530000=5.3\times 10^5$.

The exponent records the scale while the significand $a$ records the reported digits. Multiplication adds the exponents, so $(2\times 10^3)(3\times 10^{-5})=6\times 10^{-2}$. The displayed precision must still reflect the accuracy of the underlying measurement; rewriting a value does not add information. The number zero is written separately because the normalized form excludes it.

The digit sum of a nonnegative integer in a specified radix is the sum of its numerical digits. Ifwrite $s_b(n)={\sum }_j{d}_j$, with $s_b(0)=0$. For example, $s_{10}(347)=3+4+7=14$. In binary representation, the digit sum is the Hamming weight of the digit string.

Since $b\equiv 1(\mathrm{mod}b-1)$, the positional expansion givesIn decimal representation, this yields the familiar divisibility tests by $3$ and $9$: the integer and its digit sum have the same remainder modulo either. For $347$, both $347$ and $14$ leave remainder $5$ modulo $9$. The argument explains the rule from modular arithmetic, rather than relying on a pattern noticed in examples.

polyploidy combining chromosome complements derived from different species.

A linear phosphazene polymer with two chlorine substituents per phosphorus, used as a precursor for macromolecular nucleophilic substitution.

A phosphazene polymer with two ethoxy substituents at each phosphorus, commonly prepared by alkoxide substitution of linear polydichlorophosphazene.

A polynomial quotient ring is a quotient ring of $R[X_1,\dots ,X_n]$ by an ideal of relations. Two polynomials define the same class exactly when their difference belongs to that ideal. Arithmetic is ordinary polynomial arithmetic followed by imposing the relations.

For a field $k$ and monic $f\in k[X]$ of degree $d$, every element of $k[X]/(f)$ has a unique representative of degree below $d$. The classes of $1,X,\dots ,X^{d-1}$ form a basis. For example, $\mathbb{R}[X]/(X^2+1)$ is isomorphic to $\mathbb{C}$, with the class of $X$ playing the role of $i$. The quotient is a field exactly when $f$ is irreducible; a relation alone does not ensure this.

Kronecker substitution encodes a polynomial with bounded nonnegative integer coefficients as one large integer by evaluating it at a sufficiently large base. Integer multiplication then gives the polynomial product, provided the base exceeds every output coefficient so carries cannot mix coefficients.

Carryless multiplication treats binary digits as coefficients of a polynomial over the field with two elements. Products are formed by polynomial multiplication, with exclusive or combining equal-degree terms instead of carrying to the next binary position.

The convolution of coefficient sequences $a_i,b_i$ has coefficients $c_n={\sum }_{i+j=n}{a}_i{b}_j$. These are exactly the coefficients of the product of the corresponding polynomials.

An amorphous nonmolecular form of carbon dioxide. It is distinct from molecular dry ice and from assuming a unique crystalline polymorph.^[ref]

For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_c{R}^2\sim 1$. A chain contains $N$ monomers with typical occupied area ${\ell }^2$, so its overlap area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$. With the planar good-solvent exponent $\nu =3/4$ from the two-dimensional Flory approximation, this gives ${\varphi }^{\ast }\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.

The end-to-end distance of a polymer chain with $N$ fixed-length links of length $a$ cannot exceed its polymer contour length $Na$. A Gaussian chain spring has no such bound, so it eventually becomes an unphysical extrapolation. For a three-dimensional freely jointed chain under force $f$, the mean extension along the force is $Na[\coth x-1/x]$, where $x=fa/(k_{B}T)$. Its small-$x$ limit is $Nf{a}^2/(3k_{B}T)$, and its large-$x$ limit approaches $Na$ from below.