Repunits whose digits in base $b$ are all $b-1$
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Positive integers whose base-$b$ representation contains only the digit $1$ are called repunits in that base. But what about positive integers whose base-$b$ representation contains only the digit $b-1$?
For instance, how would one call the base-$20$ number whose representation in that base is
$$19cdot19cdot19cdot19cdot19_{20}?$$
Is there a special name for this kind of number? I know repunits are useful in many number-theoretical contexts, but what about such numbers?
elementary-number-theory number-systems
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
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add a comment |
$begingroup$
Positive integers whose base-$b$ representation contains only the digit $1$ are called repunits in that base. But what about positive integers whose base-$b$ representation contains only the digit $b-1$?
For instance, how would one call the base-$20$ number whose representation in that base is
$$19cdot19cdot19cdot19cdot19_{20}?$$
Is there a special name for this kind of number? I know repunits are useful in many number-theoretical contexts, but what about such numbers?
elementary-number-theory number-systems
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
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1
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You mean like 99999 in base 10?
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– qwr
Dec 18 '18 at 17:30
2
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Probably not. They can be written as $b^{k}-1.$
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– Thomas Andrews
Dec 18 '18 at 17:30
add a comment |
$begingroup$
Positive integers whose base-$b$ representation contains only the digit $1$ are called repunits in that base. But what about positive integers whose base-$b$ representation contains only the digit $b-1$?
For instance, how would one call the base-$20$ number whose representation in that base is
$$19cdot19cdot19cdot19cdot19_{20}?$$
Is there a special name for this kind of number? I know repunits are useful in many number-theoretical contexts, but what about such numbers?
elementary-number-theory number-systems
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
$endgroup$
Positive integers whose base-$b$ representation contains only the digit $1$ are called repunits in that base. But what about positive integers whose base-$b$ representation contains only the digit $b-1$?
For instance, how would one call the base-$20$ number whose representation in that base is
$$19cdot19cdot19cdot19cdot19_{20}?$$
Is there a special name for this kind of number? I know repunits are useful in many number-theoretical contexts, but what about such numbers?
elementary-number-theory number-systems
elementary-number-theory number-systems
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
asked Dec 18 '18 at 17:27
KlangenKlangen
1,69811334
1,69811334
1
$begingroup$
You mean like 99999 in base 10?
$endgroup$
– qwr
Dec 18 '18 at 17:30
2
$begingroup$
Probably not. They can be written as $b^{k}-1.$
$endgroup$
– Thomas Andrews
Dec 18 '18 at 17:30
add a comment |
1
$begingroup$
You mean like 99999 in base 10?
$endgroup$
– qwr
Dec 18 '18 at 17:30
2
$begingroup$
Probably not. They can be written as $b^{k}-1.$
$endgroup$
– Thomas Andrews
Dec 18 '18 at 17:30
1
1
$begingroup$
You mean like 99999 in base 10?
$endgroup$
– qwr
Dec 18 '18 at 17:30
$begingroup$
You mean like 99999 in base 10?
$endgroup$
– qwr
Dec 18 '18 at 17:30
2
2
$begingroup$
Probably not. They can be written as $b^{k}-1.$
$endgroup$
– Thomas Andrews
Dec 18 '18 at 17:30
$begingroup$
Probably not. They can be written as $b^{k}-1.$
$endgroup$
– Thomas Andrews
Dec 18 '18 at 17:30
add a comment |
1 Answer
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While there is no name, to my knowledge, for the general case, they are referred to as Mersenne numbers for base 2.
answered Dec 18 '18 at 17:38
RJJBRJJB
132
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$begingroup$
While there is no name, to my knowledge, for the general case, they are referred to as Mersenne numbers for base 2.
answered Dec 18 '18 at 17:38
RJJBRJJB
132
$endgroup$
add a comment |
$begingroup$
While there is no name, to my knowledge, for the general case, they are referred to as Mersenne numbers for base 2.
answered Dec 18 '18 at 17:38
RJJBRJJB
132
$endgroup$
add a comment |
$begingroup$
While there is no name, to my knowledge, for the general case, they are referred to as Mersenne numbers for base 2.
answered Dec 18 '18 at 17:38
RJJBRJJB
132
$endgroup$
While there is no name, to my knowledge, for the general case, they are referred to as Mersenne numbers for base 2.
answered Dec 18 '18 at 17:38
RJJBRJJB
132
answered Dec 18 '18 at 17:38
RJJBRJJB
132
answered Dec 18 '18 at 17:38
RJJBRJJB
132
answered Dec 18 '18 at 17:38
RJJBRJJB
132
132
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1
$begingroup$
You mean like 99999 in base 10?
$endgroup$
– qwr
Dec 18 '18 at 17:30
2
$begingroup$
Probably not. They can be written as $b^{k}-1.$
$endgroup$
– Thomas Andrews
Dec 18 '18 at 17:30