Gelfand’s question

Jaap Top

Bernoulli Institute & DIAMANT

June 25th, 2018

(PAM symposium, Groningen)

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Israel M. Gel’fand (1913–2009)

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Note that mathworld already provides the answer to the first

question: (does 9 occur as the most significant decimal digit of

some 2n?). Here is a small table.

n 1 2 3 4 5 6 7 8 9 102n 2 4 8 16 32 64 128 256 512 1024

11 12 13 14 15 16 172048 4096 8192 16384 32768 65536 131072

18 19 20 21 22262144 524288 1048576 2097152 4194304

So, not with 1 ≤ n ≤ 22.

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Staring at the table, one might be tempted to think that the

array of blue digits is periodic with period 10.

In other words: do 2n and 2n+10 have the same most significant

decimal digit?

Example: 21 and 211 and 221 and 231 and 241 and 251 and 261

all have leftmost decimal digit 2.

The same for 271, 281, . . . ,2161,2171

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But then

2181 =

3064991081731777716716694054300618367237478244367204352

and 2301 has most significant decimal digit 4,

and 2391 even has leftmost decimal digit 5.

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Concerning the occurrence of a leftmost decimal digit 7 or 9,

246 = 70368744177664

253 = 9007199254740992

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9

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So how about the remaining questions 2) and 3) of Gel’fand:

2) Does n > 1 exist such that the array of leftmost decimal digits

of (2n,3n,4n, . . . ,8n,9n) equals (2,3,4,5,6,7,8,9)?

3) Do n > 0 and ` ∈ {1,2,3, . . . ,8,9} exist such that the ar-

ray of leftmost decimal digits of (2n,3n,4n, . . . ,8n,9n) equals

(`, `, `, `, `, `, `, `)?

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Blogger / mathematician John D. Cook used the summer of2013 to verify:

For no n with 1 < n < 1010 is (2,3,4,5,6,7,8,9), or is any(`, `, `, `, `, `, `, `) the array of leftmost digits of (2n,3n,4n, . . . ,9n).

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Answering Gel’fand’s question turns out to be surprisingly simple!

It was done by Jaap Eising as a small part of his bachelor’s

project (2013, Groningen):

Questions 2) and 3) have the same answer: NO.

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Jaap Eising

(and his fellow board members of FMF, 2014)

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To answer question 2): write

2n = αn × 10m

for a nonnegative integer m, and 1 < αn < 10.

With this notation, the leading decimal digit of 2n equals bαnc.

Since 2n = 10τn with τ = log(2)/ log(10), we have m = bτnc and

αn = 10τn−bτnc.

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We treat 5n in a similar way: so 5n = γn × 10k where k is a

nonnegative integer and 1 < γn < 10.

As 2n × 5n = 10n, we have αn × γn = 10.

Now suppose that for n > 1 it holds that bαnc = 2 and also

bγnc = 5.

That means 2 ≤ αn < 3 and 5 ≤ γn < 6.

Of course αn 6= 2, because 2n 6= 2×10m (remember that n > 1).

So 10 = αn × γn > 10, a contradiction!.

Conclusion: there is no n > 1 for which the leading decimal digits

of 2n and 5n are 2 and 5, respectively.

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A similar argument works for Question 3):

if the leading decimal digit of each of 2n to 9n were the same,

then using αn × γn = 10 and bαnc = bγnc,

it follows that 3 < αn, γn < 4.

However, in this case 4n = 2n×2n = α2n×102m will have leading

decimal digit either 9 or 1, because 9 < α2n < 16. As we saw that

2n must have leading decimal digit equal to 3,

we conclude: all of 2n, . . . ,9n having the same leading decimal

digit, does not occur!

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We return to question 1).

Here 2n = αn × 10m, and with τ = log(2)/ log(10) we have

m = bτnc and αn = 10τn−bτnc.

So, if we want 2n to have a certain leading decimal digit (say `),

then we want ` ≤ αn < `+ 1,

which means log(`)/ log(10) ≤ τn− bτnc < log(`+ 1)/ log(10).

Leopold Kronecker showed in 1884 that for any σ ∈ R\Q it holds

that the numbers (σn− bσnc)n≥1 are dense in the interval [0,1].

Piers Bohl, Wac law Sierpinski, and Hermann Weyl independently

proved in 1909-1910 a stronger result:

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Thm. For every σ ∈ R \ Q and every integrable f : [0,1] → C itholds that ∫ 1

0f(t) dt = lim

N→∞1

N

N∑n=1

f(σn− bσnc).

Applying this to

f(t) :=

{1 if log(`)/ log(10) ≤ t < log(`+ 1)/ log(10);0 otherwise,

shows that

limN→∞

#{n ≤ N : 2n has leading digit `}N

=

log(`+ 1)/ log(10)− log(`)/ log(10) = log(1 + 1`)/ log(10).

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So this tells which ‘proportion’ of all n yield leading digit of 2n

equal to `.

Note that the answer is the same if we consider 3n, or 4n, or 5n,

et cetera.

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Details and more can be found in a paper published in 2015 in

the American Mathematical Monthly:

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