Full number

A full number is a positive integer that is divided entirely by the square of each prime divisor .

Equivalent definition: a number represented as $a^{2}b^{3}$ ${\ displaystyle a ^ {2} b ^ {3}}$ ${\ displaystyle a ^ {2} b ^ {3}}$ where $a$ ${\ displaystyle a}$ $a$ and $b$ ${\ displaystyle b}$ $b$ Are positive integers.

Full numbers are systematically studied by Pal Erdös and Györök Szérés , given by Solomon Golomb .

List of full numbers between 1 and 1000 ^[1] :

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.

Equivalence of Two Definitions

If a $m=a^{2}b^{3}$ ${\ displaystyle m = a ^ {2} b ^ {3}}$ $m=a^{2}b^{3}$ , then any simple decomposition $a$ ${\ displaystyle a}$ $a$ comes in twice, and the incoming $b$ ${\ displaystyle b}$ $b$ - at least three times; so any easy decomposition $m$ ${\ displaystyle m}$ $m$ included no less than a square .

On the other hand, let $m$ ${\ displaystyle m}$ $m$ - full number with expansion

m=\prod p_{i}^{\alpha _{i}}

{\ displaystyle m = \ prod p_ {i} ^ {\ alpha _ {i}}}

m=\prod p_{i}^{\alpha _{i}}

,

where each $\alpha _{i}\geq 2$ ${\ displaystyle \ alpha _ {i} \ geq 2}$ $\alpha _{i}\geq 2$ . Define $\gamma _{i}$ ${\ displaystyle \ gamma _ {i}}$ $\gamma _{i}$ equal to three if $\alpha _{i}$ ${\ displaystyle \ alpha _ {i}}$ $\alpha_i$ odd, and zero otherwise, and define $\beta _{i}=\alpha _{i}-\gamma _{i}$ ${\ displaystyle \ beta _ {i} = \ alpha _ {i} - \ gamma _ {i}}$ $\beta _{i}=\alpha _{i}-\gamma _{i}$ . Then all the values $\beta _{i}$ ${\ displaystyle \ beta _ {i}}$ $\beta _{i}$ are non-negative even integers and all values $\gamma _{i}$ ${\ displaystyle \ gamma _ {i}}$ $\gamma _{i}$ either zero or three, so:

m=(\prod p_{i}^{\beta _{i}})(\prod p_{i}^{\gamma _{i}})=(\prod p_{i}^{\beta _{i}/2})^{2}(\prod p_{i}^{\gamma _{i}/3})^{3}

{\ displaystyle m = (\ prod p_ {i} ^ {\ beta _ {i}}) (\ prod p_ {i} ^ {\ gamma _ {i}}) = (\ prod p_ {i} ^ {\ beta _ {i} / 2}) ^ {2} (\ prod p_ {i} ^ {\ gamma _ {i} / 3}) ^ {3}}

m=(\prod p_{i}^{\beta _{i}})(\prod p_{i}^{\gamma _{i}})=(\prod p_{i}^{\beta _{i}/2})^{2}(\prod p_{i}^{\gamma _{i}/3})^{3}

gives the desired representation $m$ ${\ displaystyle m}$ $m$ as a product of a square and a cube.

In other words, for a given expansion of the number $m$ ${\ displaystyle m}$ $m$ can be taken as $b$ ${\ displaystyle b}$ $b$ the product of prime factors in the expansion with odd powers (if there are none, then 1). Insofar as $m$ ${\ displaystyle m}$ $m$ - full, every simple factor included in the decomposition with an odd degree has a degree of at least 3, so $m/b^{3}$ ${\ displaystyle m / b ^ {3}}$ $m/b^{3}$ is the whole. Now every prime factor $m/b^{3}$ ${\ displaystyle m / b ^ {3}}$ $m/b^{3}$ has an even degree so that $m/b^{3}$ ${\ displaystyle m / b ^ {3}}$ $m/b^{3}$ Is a full square, we denote it as $a^{2}$ ${\ displaystyle a ^ {2}}$ $a^2$ ; and it turns out $m=a^{2}b^{3}$ ${\ displaystyle m = a ^ {2} b ^ {3}}$ $m=a^{2}b^{3}$ . For example:

m=21600=2^{5}\times 3^{3}\times 5^{2}

{\ displaystyle m = 21600 = 2 ^ {5} \ times 3 ^ {3} \ times 5 ^ {2}}

m=21600=2^{5}\times 3^{3}\times 5^{2}

,

b=2\times 3=6

{\ displaystyle b = 2 \ times 3 = 6}

b=2\times 3=6

,

a={\sqrt {\frac {m}{b^{3}}}}={\sqrt {2^{2}\times 5^{2}}}=10

{\ displaystyle a = {\ sqrt {\ frac {m} {b ^ {3}}}} = {\ sqrt {2 ^ {2} \ times 5 ^ {2}}} = 10}

a={\sqrt {\frac {m}{b^{3}}}}={\sqrt {2^{2}\times 5^{2}}}=10

,

m=a^{2}b^{3}=10^{2}\times 6^{3}

{\ displaystyle m = a ^ {2} b ^ {3} = 10 ^ {2} \ times 6 ^ {3}}

m=a^{2}b^{3}=10^{2}\times 6^{3}

.

Mathematical Properties

The sum of the inverse of the full numbers converges:

\prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315}{2\pi ^{4}}}\zeta (3)

{\ displaystyle \ prod _ {p} \ left (1 + {\ frac {1} {p (p-1)}} \ right) = {\ frac {\ zeta (2) \ zeta (3)} {\ zeta (6)}} = {\ frac {315} {2 \ pi ^ {4}}} \ zeta (3)}

,

Where $p$ ${\ displaystyle p}$ - bypasses all prime numbers, $\zeta (s)$ ${\ displaystyle \ zeta (s)}$ Is the Riemann zeta function , and $\zeta (3)$ ${\ displaystyle \ zeta (3)}$ - Aperi constant (Golomb, 1970).

Let be $k(x)$ ${\ displaystyle k (x)}$ means the number of full times in the range $[1,x]$ ${\ displaystyle [1, x]}$ . Then $k(x)$ ${\ displaystyle k (x)}$ in proportion to the square root of $x$ ${\ displaystyle x}$ . More precisely:

cx^{1/2}-3x^{1/3}\leq k(x)\leq cx^{1/2},c=\zeta (3/2)/\zeta (3)=2,173\cdots

{\ displaystyle cx ^ {1/2} -3x ^ {1/3} \ leq k (x) \ leq cx ^ {1/2}, c = \ zeta (3/2) / \ zeta (3) = 2,173 \ cdots}

^[2] .

The two smallest consecutive full numbers are 8 and 9. Since the Pell equation $x^{2}-8y^{2}=1$ ${\ displaystyle x ^ {2} -8y ^ {2} = 1}$ has an infinite number of solutions, then there is an infinite number of pairs of consecutive full numbers ^[2] ; More generally, one can find consecutive full numbers by finding a solution to an equation similar to the Pell equation, $x^{2}-ny^{2}=\pm 1$ ${\ displaystyle x ^ {2} -ny ^ {2} = \ pm 1}$ for any cube $n$ ${\ displaystyle n}$ . However, one of the full numbers in the pair thus obtained should be a square. According to Guy, Erdös asked the question whether the number of pairs of full times similar to $(23^{3},2^{3}\cdot 3^{2}\cdot 13^{2})$ ${\ displaystyle (23 ^ {3}, 2 ^ {3} \ cdot 3 ^ {2} \ cdot 13 ^ {2})}$ in which none of the numbers in the pair is a square. Yaroslav Vroblevsky showed that, on the contrary, there are infinitely many such pairs, showing that $3^{3}c^{2}+1=7^{3}d^{2}$ ${\ displaystyle 3 ^ {3} c ^ {2} + 1 = 7 ^ {3} d ^ {2}}$ has infinitely many solutions.

According to the Erdшаs – Mollin – Walsh hypothesis , there are no three consecutive full numbers.

Sum and difference of full numbers

Any odd number can be represented as the difference of two consecutive squares:

(k+1)^{2}=k^{2}+2k+1\Rightarrow (k+1)^{2}-k^{2}=2k+1

{\ displaystyle (k + 1) ^ {2} = k ^ {2} + 2k + 1 \ Rightarrow (k + 1) ^ {2} -k ^ {2} = 2k + 1}

.

In the same way, any multiple of four is representable as the difference of two numbers that differ by two: $(k+2)^{2}-k^{2}=4k+4$ ${\ displaystyle (k + 2) ^ {2} -k ^ {2} = 4k + 4}$ . However, a number divisible by two, but not four, cannot be represented as the difference of squares, that is, the question arises: which even numbers that are not divisible by 4 can be represented as the difference of two full numbers.

Golomb gave several such representations:

2 = 3 ³ - 5 ²

10 = 13 ³ - 3 ⁷

18 = 19 ² - 7 ³ = 3 ² (3 ³ - 5 ² ).

At first, it was hypothesized that the number 6 cannot be represented in this form, and Golomb suggested that there are infinitely many integers that cannot be represented as the difference of two full numbers. However, Narkivich found that there are infinitely many ways to represent the number 6, for example

6 = 5 ⁴ 7 ³ - 463 ² ,

and McDaniel ^[3] showed that any number has an infinite number of such representations.

Erdös hypothesized that any sufficiently large integer is the sum of a maximum of three full numbers. The hypothesis was proved by Roger Heath-Brown ^[4] .

Summary

$k$ ${\ displaystyle k}$ - full numbers - numbers in the expansion of which prime numbers are included with a degree of at least $k$ ${\ displaystyle k}$ .

$(2^{k+1}-1)^{k}$ ${\ displaystyle (2 ^ {k + 1} -1) ^ {k}}$ , $2^{k}(2^{k+1}-1)^{k}$ ${\ displaystyle 2 ^ {k} (2 ^ {k + 1} -1) ^ {k}}$ , $(2^{k+1}-1)^{k+1}$ ${\ displaystyle (2 ^ {k + 1} -1) ^ {k + 1}}$ are $k$ ${\ displaystyle k}$ -fold in arithmetic progression .

Moreover, if $a_{1},a_{2},\dots ,a_{s}$ ${\ displaystyle a_ {1}, a_ {2}, \ dots, a_ {s}}$ are $k$ ${\ displaystyle k}$ -fold in arithmetic progression with a difference $d$ ${\ displaystyle d}$ then:

(a_{1}+d)^{k},a_{2}(a_{s}+d)^{k},\dots ,a_{s}(a_{s}+d)^{k},a_{s}(a_{s}+d)^{k+1}

{\ displaystyle (a_ {1} + d) ^ {k}, a_ {2} (a_ {s} + d) ^ {k}, \ dots, a_ {s} (a_ {s} + d) ^ { k}, a_ {s} (a_ {s} + d) ^ {k + 1}}

are $k$ ${\ displaystyle k}$ -fold numbers in arithmetic progression.

For $k$ ${\ displaystyle k}$ - full numbers takes place:

a^{k}(a^{l}+\dots +1)^{k}+a^{k+1}(a^{l}+\dots +1)+\dots +a^{k+l}(a^{l}+\dots +1)=a^{k}(a^{l}+\dots +1)^{k+1}

{\ displaystyle a ^ {k} (a ^ {l} + \ dots +1) ^ {k} + a ^ {k + 1} (a ^ {l} + \ dots +1) + \ dots + a ^ {k + l} (a ^ {l} + \ dots +1) = a ^ {k} (a ^ {l} + \ dots +1) ^ {k + 1}}

.

This equality gives infinitely many sets of length $l+1$ ${\ displaystyle l + 1}$ $k$ ${\ displaystyle k}$ - full numbers whose sums are also $k$ ${\ displaystyle k}$ - full time. Nitaj ^[5] showed that there are infinitely many solutions to the equation $x+y=z$ ${\ displaystyle x + y = z}$ among coprime 3-fold numbers. Cohn ^[6] constructed an infinite family of solutions of the equation $x+y=z$ ${\ displaystyle x + y = z}$ among coprime 3-fold numbers: three

X=9712247684771506604963490444281

{\ displaystyle X = 9712247684771506604963490444281}

,

Y=32295800804958334401937923416351

{\ displaystyle Y = 32295800804958334401937923416351}

,

Z=27474621855216870941749052236511

{\ displaystyle Z = 27474621855216870941749052236511}

is a solution to the equation $32X^{3}+49Y^{3}=81Z^{3}$ ${\ displaystyle 32X ^ {3} + 49Y ^ {3} = 81Z ^ {3}}$ . It is possible to construct another solution by putting $X'=X(49Y^{3}+81Z^{3}),Y'=-Y(32X^{3}+81Z^{3}),Z'=Z(32X^{3}-49Y^{3})$ ${\ displaystyle X '= X (49Y ^ {3} + 81Z ^ {3}), Y' = - Y (32X ^ {3} + 81Z ^ {3}), Z '= Z (32X ^ {3} -49Y ^ {3})}$ and removing the common factor.

Notes

↑ sequence A001694 in OEIS
↑ ¹ ² Golomb, 1970 .
↑ McDaniel, 1982 .
↑ Heath-Brown, 1988 .
↑ Nitaj, 1995 .
↑ Cohn, 1998 .

Literature

Cohn, JHE A conjecture of Erdős on 3-powerful numbers // Math. Comp. - 1998. - T. 67 , no. 221 . - S. 439-440 . - DOI : 10.1090 / S0025-5718-98-00881-3 .
Pál Erdős, György Szekeres. Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem // Acta Litt. Sci. Szeged. - 1934. - No. 7 . - S. 95-102 .
Solomon W. Golomb. Powerful numbers // American Mathematical Monthly . - 1970. - T. 77 , No. 8 . - S. 848–852 . - DOI : 10.2307 / 2317020 .
Richard K. Guy. Section B16 // Unsolved Problems in Number Theory, 3rd edition. - Springer-Verlag, 2004. - ISBN 0-387-20860-7 .
Roger Heath-Brown. Ternary quadratic forms and sums of three square-full numbers. - Boston: Birkhäuser, 1988 .-- pp. 137-163. - (Séminaire de Théorie des Nombres, Paris, 1986-7).
Roger Heath-Brown. Sums of three square-full numbers. - Colloq. Math. Soc. János Bolyai, no. 51, 1990 .-- S. 163-171. - (Number Theory, I (Budapest, 1987)).
Wayne L. McDaniel. Representations of every integer as the difference of powerful numbers // Fibonacci Quarterly . - 1982. - No. 20 . - S. 85–87 .
Abderrahmane Nitaj. On a conjecture of Erdős on 3-powerful numbers // Bull. London Math. Soc. . - 1995. - T. 4 , No. 27 . - S. 317-318 . - DOI : 10.1112 / blms / 27.4.317 .

Links

Weisstein, Eric W. Powerful number on the Wolfram MathWorld website.
The abc conjecture

[1] sequence A001694 in OEIS

[_bc3a604447efc206-2] ¹ ² Golomb, 1970 .

[_4a1a429c4b496946-3] McDaniel, 1982 .

[_9840a20f7e2c995a-4] Heath-Brown, 1988 .

[_89bb38ebe22434db-5] Nitaj, 1995 .

[_89916eb66202be98-6] Cohn, 1998 .