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In number theory, themultiplicative digital root of anatural number${\displaystyle n}$ in a givennumber base${\displaystyle b}$ is found bymultiplying thedigits of${\displaystyle n}$ together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of${\displaystyle n}$.^[1][2] The multiplicative digital root for the first few positive integers are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. (sequenceA031347 in theOEIS)

Multiplicative digital roots are the multiplicative equivalent ofdigital roots, with a major difference being that for natural numbers in base${\displaystyle b=10}$, the multiplicative digital roots can be 0 to 9, whereas digital roots can only be 1 to 9.

Let${\displaystyle n}$ be a natural number. We define thedigit product for base${\displaystyle b>1}$${\displaystyle F_b:\mathbb{N}\to \mathbb{N}}$ to be the following:

${\displaystyle F_b(n)=\prod _{i=0}^{k-1}{d}_i}$

where${\displaystyle k=\lfloor {\log }_{b}n\rfloor +1}$ is the number of digits in the number in base${\displaystyle b}$, and

${\displaystyle d_i=\frac{nmod{b}^{i+1}-n{modb}^i}{b^i}}$

is the value of each digit of the number. A natural number${\displaystyle n}$ is amultiplicative digital root if it is afixed point for${\displaystyle F_b}$, which occurs if${\displaystyle F_b(n)=n}$.

For example, in base${\displaystyle b=10}$, 0 is the multiplicative digital root of 9876, as

${\displaystyle F_{10}(9876)=(9)(8)(7)(6)=3024}$

${\displaystyle F_{10}(3024)=(3)(0)(2)(4)=0}$

${\displaystyle F_{10}(0)=0}$

All natural numbers${\displaystyle n}$ arepreperiodic points for${\displaystyle F_b}$, regardless of the base. This is because if${\displaystyle n\ge b}$, then

${\displaystyle n=\sum _{i=0}^{k-1}{d}_i{b}^i}$

and therefore

If, then trivially

${\displaystyle F_b(n)=n}$

Therefore, the only possible multiplicative digital roots are the natural numbers, and there are no cycles other than the fixed points of.

The number of iterations${\displaystyle i}$ needed for${\displaystyle F_b^i(n)}$ to reach a fixed point is themultiplicativepersistence of${\displaystyle n}$. The multiplicative persistence is undefined if it never reaches a fixed point.

Inbase 10, it is conjectured that there is no number with a multiplicative persistence${\displaystyle i>11}$: this is known to be true for numbers${\displaystyle n\le {10}^{20585}}$.^[3][4] The smallest numbers with persistence 0, 1, ... are:

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. (sequenceA003001 in theOEIS)

The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.Based on these restrictions, the number of candidates for${\displaystyle k}$-digit numbers with record-breaking persistence is only proportional to the square of${\displaystyle k}$, a tiny fraction of all possible${\displaystyle k}$-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.^[3]

The multiplicative digital root can be extended to the negative integers by use of asigned-digit representation to represent each integer.

The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences inPython.

defdigit_product(x:int,b:int)->int:ifx==0:return0total=1whilex>1:ifx%b==0:return0ifx%b>1:total=total*(x%b)x=x//breturntotaldefmultiplicative_digital_root(x:int,b:int)->int:seen=[]whilexnotinseen:seen.append(x)x=digit_product(x,b)returnxdefmultiplicative_persistence(x:int,b:int)->int:seen=[]whilexnotinseen:seen.append(x)x=digit_product(x,b)returnlen(seen)-1

• Arithmetic dynamics
• Digit sum
• Digital root
• Sum-product number

• Guy, Richard K. (2004).Unsolved problems in number theory (3rd ed.).Springer-Verlag. pp. 398–399.ISBN 978-0-387-20860-2.Zbl 1058.11001.

• What's special about 277777788888899? - Numberphile onYouTube (Mar 21, 2019)

• Algebra
• Arithmetic dynamics
• Integer sequences
• Base-dependent integer sequences
• Number theory

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