Glossary Example

It is assumed that you are familiar with the following concepts:

Number Base (Radix). Let the integer r ≥ 2 be a number base (or radix), and let the integers b and n be greater than or equal to 0. We can represent an arbitrary integer a as

[1]     a = br + n     [4]

Digit. Let n be a digit of base r, with 0 < nr. The digit “0” is used to signify that an arbitrary integer a is congruent with r, i.e., a ≡ 0 (mod r) [5]. We will ignore the case where a = 0. There must be r numerals to represent base r in standard positional notation. [6] In general (since the start of 2011), I use the term “digit” to mean an integer n, “numeral” to mean the symbol and name of n, and “place-value” to signify a digit in a number such as “2011”, wherein the most significant place-value is 2. (See Wolfram MathWorld’s definitions of Digit and Numeral).

Prime Decomposition. Each positive nonzero integer can be uniquely represented as the product of powers of primes [7]. Let the prime p be a divisor of r, i.e. p | r, and let the prime q be coprime to r, i.e. qr. We can write the standard form for prime power decomposition of r as:

[2]     r = p1ρ1 · p2ρ2 · … · pkρk
with p1 < p2 < … pk

Unit. A unit is the digit 1, neither prime nor composite, both a divisor of and coprime to r. Units appear in purple in the digit map below.

Figure 1: Decimal Unit Digits

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Divisor. A divisor is an integer d that, multiplied by a second integer d′, produces r [8]:

[3]     r = d · d′.

Let the integer k ≥ 1 represent the number of distinct prime divisors p of r. Let the integers ρ > 0 and δ ≥ 0 be the exponents of any prime factor of the number base r and its divisors d, respectively. The divisor d of r has the prime decomposition

[4]     d = p1δ1 · p2δ2 · … · pkδk
with p1 < p2 < … pk and with no δ > ρ

Thus, no divisor of the number base can have an exponent of any prime factor exceed that of the corresponding prime factor in the number base itself. The divisor digits {0, 1} (the divisors {1, r}) are divisors of each base r, and are called “trivial divisors” [9]. The divisor is a kind of regular number. See the MathWorld definition of divisor. Divisors appear in red (purple for the unit divisor) below.

Figure 2: Decimal Divisor Digits

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Regular number. Let the integer g have the form shown in Formula 4, without the restriction that “no δ > ρ[10, 11]. The regular number g thus is a product solely of prime divisors p of base r, with any exponent δ > 0. The decimal regular numbers include products g of any positive power of 2 and any positive power of 5: {1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, …}.

Regular digit. A regular digit is a regular number gr, with the digit “0” signifying congruence with r. The decimal regular digits are {0, 1, 2, 4, 5, 8} whilst the regular numbers less than or equal to r are {1, 2, 4, 5, 8, 10}. Thus the set of divisor digits is a subset of the set of regular digits of r. Regular digits appear in red, purple, and orange below.

Figure 3: Decimal Regular Digits

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“Semidivisor”. I call a composite nondivisor regular digit gr a “semidivisor”. The decimal semidivisors are {4, 8}. Semidivisors are one of two types of “neutral digit”. Semidivisors must be composite digits of composite bases

r. Semidivisors appear in orange below. (Another term I considered was “quasidivisor”, but selected “semidivisor” in symphony with the term “semitotative”, the other kind of neutral digit.)

Figure 4: Decimal Semidivisors

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Semidivisors exist for all composite number bases r ≥ 6.

Totative. Let the integer t < r be such that gcd(t, r) = 1; thus t is coprime to base r, i.e., tr [12, 13, 14]. Then t has the prime decomposition

[5]     t = q1ρ1 · q2ρ2 · … · qkρk
with q1 < q2 < … qk

The totative t and the number base r have no factor in common but 1; t is “out of phase” with r. See the MathWorld definitions of relatively prime and totative. Totatives appear in gray, light blue, and purple (for the unit totative) below. They are colored light green in Figures 6 and 7.

Figure 5: Decimal Totatives

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Neighbor-Factor. Let the integers α = (r + 1) and ω = (r − 1). A neighbor-factor is a divisor of either α or ω [15, 16]. The digit 2 in odd bases r is a divisor of both α or ω. The neighbor-factors are coprime to r, since 2 is the smallest prime, and the difference between r and r ± 1 is by definition less than 2.

Figure 5: Factors of Decimal Neighbors

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Figure 6: Factors of Tetradecimal (Base 14) Neighbors

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Figure 7: Factors of Pentadecimal (Base 15) Neighbors

0 1 2 3 4 5 6 7 8 9 a b c d e

Alpha-Factor (sometimes called “alphas” though technically only r + 1 = α). A divisor of α. Let tα be a divisor of (r + 1). Then tα cannot divide r evenly, since no prime is smaller than 2. All the prime factors of (r + 1) must be q, thus gcd(r, (r + 1)) = 1. Alpha-factors appear in light-green in Figures 6 and 7 above, and in light-purple (digit 2 base 15) in Figure 7 above.

Omega-Factor (sometimes called “omegas”) though tecnically only r − 1 = ω. A divisor of ω. Let tω be a divisor of (r − 1). Then tω cannot divide r evenly, since no prime is smaller than 2. All the prime factors of (r − 1) must be q, thus gcd(r, (r − 1)) = 1. Omega-factors appear in light-blue in Figures 5–7 above, and in light-purple (digit 2 base 15) in Figure 7 above.

Alpha/Omega-Factor. For all odd bases r, digit 2 | (r − 1) and 2 | (r + 1). Thus, digit 2 in all odd bases r is related to both α and ω. The behavior of digit 2 in odd bases is that of an ω-factor. The alpha/omega-factor 2 appears in light purple in Figure 7 above.

“Opaque” Totative. I call a totative “opaque” if it is a digit tr that does not divide either or both α = (r + 1) nor ω = (r − 1). The notion of “opacity” relates to the notion of totatives as providing little leverage for human intuitive computation. Some opaque totatives are not “opaque“ concerning divisibility tests. Totatives that inherit intuitive divisibility tests from an α- and ω-factors indeed have composite intuitive divisibility rules.

Figure 8: The Decimal Opaque Totative

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“Neutral Digit”. I call a digit that is neither a divisor nor a totative of r a “neutral digit”. Neutral digits must be composite digits. There are two types of neutral digit: the semidivisor (see above), and the semitotative. The semidivisor is a product solely of any power δ of any of the prime divisors p of base r, and is a regular number g le; r. The semitotative is a mixed product pq, with p a prime divisor of r and q a prime that is coprime to r. The semitotative is a semi-coprime integer h < r. Neutral digits appear in gold in Figure 9.

Figure 9: Decimal Neutral Digits

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“Semi-Coprime Number”. A “semi-coprime number” is an integer h that is simply the product pq of at least one prime divisor p and at least one prime q that is coprime to r [17]. The decimal semi-coprime numbers are {6, 12, 14, 15, 18, 21, 22, 24, 26, 28, 30, …}. The semitotative relationship to semi-coprime numbers is the same as the semidivisor relationship to regular numbers: these are simply the portion of the larger set that is less than or equal to r, thus able to be represented as digits of r.

“Semitotative”. Consider a composite digit h < r that is the product pq of at least one prime divisor p and at least one prime totative q of r. I call such a digit a “semitotative”. Semitotatives exist for all composite bases r that are not powers of primes p(rpρ). Thus octal is the smallest r to possess a semitotative (digit 6). For large, highly composite bases r, semitotatives abound, comprising the predominance of digits of r, but semitotatives are rare in bases 2 ≤ r≤ 20. The set of semitotatives of r are a subset of the semi-coprime numbers h. The decimal semitotative is digit 6. The hexadecimal semitotatives are digits {6, 10, 12, 14}.

“Opaque” Semitotative / Semi-Coprime Number. I call a semitotative or semi-coprime number “opaque” if it is the product pq with no intuitive divisibility test for q in base r.

Figure 10: The Decimal Semitotative

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“Digitmap”. Given the types of digits described above, it is possible to produce a “map” of all the relationsips of digits n with base r. See Figure 11. This uses the legend in Figure 12.

Figure 11: The Decimal Digit Map

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Figure 12: Legend

Unit, 1 | r ∧ 1 ⊥ r

Divisor, d | r

Semidivisor, g = ∏ piδ ∧ (δ > ρ) ∧ (g ¬| r) ∧ (0 < gr)

Semitotative, h = ∏ piqj ∧ (0 < h < r)

Totative (opaque), tr ∧ 0 < t < r

α-Totative, tαr ∧ , tα | (r + 1)

ω-Totative, tωr ∧ , tω | (r − 1)

α/ω-Totative, tαtω

“Digit Spectrum”. We can produce a chart similar to the digit map wherein we are simply representing the ratio of the quantity of digits of a given type to the magnitude of r. This is especially useful for large bases r.

Figure 13: The Decimal Digit Spectrum

                   

References:

See the OP for the list of books referenced in the following endnotes:

  1. Dudley 1969, Section 2, “Unique factorization”, page 10.
  2. LeVeque 1962, Chapter 1, “Foundation”, Section 1–5, “Radix representation”, pages 17–19.
  3. Hardy & Wright 2008, Chapter I, “The Series of Primes”, Section 1.2, “Prime numbers”, page 2.
  4. Jones & Jones 2005, Chapter 2, “Prime Numbers“, Section 2.1, “Prime numbers and prime-power factorisations“, page 20.
  5. Dudley 1969, Section 4, “Congruences”, page 28.
  6. Dudley 1969, Section 13, “Number in Other Bases”, page 106.
  7. Dudley 1969, Section 2, “Unique factorization”, pages 16–18
  8. Ore 1948, Chapter 5, “The aliquot parts”, page 86.
  9. Ore 1948, Chapter 2, “Properties of numbers. Division”, page 29.
  10. Ore 1948, Chapter 13, “Theory of Decimal Expansions”, page 316.
  11. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.2, “Terminating and recurring decimals”, page 142.
  12. Hardy & Wright 2008, Chapter V, “Congruences and Residues”, Section 5.1, “Highest common divisor and least common multiple”, page 58.
  13. Jones & Jones 2005, Chapter 1, “Divisibility”, Section 1.1, “Divisors”, page 10.
  14. LeVeque 1962, Chapter 2, “The Euclidean Algorithm and Its Consequences”, Section 2–2, “The Euclidean algorithm and greatest common divisor”, page 24.
  15. LeVeque 1962, Chapter 1, “Foundation”, Section 1–3, “Proofs by Induction“, pages 11–12.
  16. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.6, “Tests for divisibility”, pages 142-143, 147.
  17. Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.2, “Terminating and recurring decimals”, pages 142-145.

 

This page last modified Monday 22 October 2012.