Unsolved Problems in Number Theory

Guy, R. K. (2004). Unsolved Problems in Number Theory. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2004

DOI

10.1007/978-0-387-26677-0

ISBN

9781441919281

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-0-387-26677-0

Cited by

11 citations across 11 artifacts.

Domain-specific

• Arithmetic number
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Deficient number
  • Generalizations to ideals, polynomials, or other divisor theories require separately defined sums and order relations.
• Fortunate number
  • The entry states the exact integer sequence and known elementary consequences; computation is evidence, not proof that every term is prime.
• Generalized taxicab number
  • Parameter order and representation conventions must be stated, existence is unknown for many tuples and allowing negative terms defines different families.
• Golomb sequence
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Highly cototient number
  • … distinction between constitutive identity and a convenient observable transfers from To certify a term, enumerate or prove all solutions of x-phi(x)=k and compare their count with every prior k. to A computational list states search bounds and proof of preimage completeness rather than assuming no larger x maps to k..
• Noncototient
  • … identity and a convenient observable transfers from An integer is certified noncototient only after proving that the cototient equation has no solution, not merely failing to find one computationally. to A search distinguishes verified range absence below a complete bound from conjectures about all members or parity..
• Perfect ruler
  • … within that domain but bounded by the need for for every integer k in the declared range 1 through m there is exactly one ordered pair of marks with positive difference k. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Persistence of a number
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Quasiperfect number
  • … is no known concrete quasiperfect integer, so the defining equation itself—not a displayed member—is the canonical example frame. to Theorems show that any quasiperfect number would have to be an odd square exceeding a very large lower bound and containing several distinct prime factors. demonstrates that continuity.
• Untouchable Number
  • It does not classify neighboring values, nor does it settle the odd-untouchable problem; those require other preimages or exclusion arguments.

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