Powerful Number
An Integer 
such that if 
, then 
, is called a powerful number. The first few are 1, 4, 8, 9, 16, 25, 27,32, 36, 49, ... (Sloane's A001694). Powerful numbers are always of the form 
for 
.
Not every Natural Number is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently largeNatural Number is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerfulnumbers, but Erdös has conjectured that there do not exist three consecutive powerful numbers. The Conjecture thatthere are no powerful number triples implies that there are infinitely many Wieferich primes (Granville 1986, Vardi 1991).
A separate usage of the term powerful number is for numbers which are the sums of the positive powers of their digits.The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (Sloane's A007532).
References
Granville, A. ``Powerful Numbers and Fermat's Last Theorem.'' C. R. Math. Rep. Acad. Sci. Canada 8, 215-218, 1986. Guy, R. K. ``Powerful Numbers.'' §B16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 67-73, 1994. Heath-Brown, D. R. ``Ternary Quadratic Forms and Sums of Three Square-Full Numbers.'' In Séminaire de Theorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhäuser, pp. 137-163, 1988. Ribenboim, P. ``Catalan's Conjecture.'' Amer. Math. Monthly 103, 529-538, 1996. Sloane, N. J. A. SequencesA001694/M3325and A007532/M0487in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59-62, 1991.
- Inversion Operation
- Inversion Radius
- Inversive Distance
- Inversive Geometry
- Inverted Funnel
- Inverted Snub Dodecadodecahedron
- Invertible Knot
- Involuntary
- Involute
- Involution (Group)
- Involution (Line)
- Involution (Operator)
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- Involution (Transformation)
- Irradiation Illusion
- Irrationality Measure
- Irrationality Sequence
- Irrational Number
- Irreducible Matrix
- Irreducible Polynomial
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- Irreducible Semiperfect Number
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- Irredundant Ramsey Number
- Irreflexive