Problems & Puzzles: Puzzles

Puzzle 668 Fortyfive

I saw recently this curio in the nice site "futility closet"

 452 = 2025 20 + 25 = 45 453 = 91125 9 + 11 + 25 = 45 454 = 4100625 4 + 10 + 06 + 25 = 45 455 = 184528125 18 + 4 + 5 + 2 + 8 + 1 + 2 + 5 = 45 456 = 8303765625 8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5 = 45

Q. Can you get another/better solution, prime or not?

Contributions came from Giovanni Resta, Jan van Delden & Hakan Summakoglu.

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Giovanni wrote:

The first base which beats 45 is 91
(991 for prime numbers, see below),
which admits decompositions for all the exponents up to 10:

91^2 => 8+2+81
91^3 => 7+5+3+5+71
91^4 => 6+8+57+4+9+6+1
91^5 => 6+2+4+0+3+21+4+51
91^6 => 5+6+7+8+6+9+2+5+2+0+41
91^7 => 5+1+6+7+6+1+0+19+3+5+7+31
91^8 => 4+7+0+2+5+2+5+2+7+6+15+15+21
91^9 => 4+2+7+9+2+9+8+0+0+1+2+9+7+8+8+4+11
91^10 => 3+8+9+4+1+6+1+1+8+1+18+10+7+4+5+4+0+1

I quickly checked all the bases up to 10^5 and the ones which set
a record are: (entry 71 964 means that 964^p can be decomposed for all the powers p from 2 to 71).

p N
2 36
6 45
10 91
50 675
68 945
71 964
107 990
114 1702
173 2728
285 4879
403 5050
443 7777
498 8938
518 9325
537 9765
543 9909
547 9918
548 9945
552 9955
567 10512
660 12222
674 12727
901 17271
1972 41149
2023 42643
2919 48790
2989 50050
3294 72612
3405 75331
3508 77778
3648 81118
3916 87571
4160 93574
4224 95121
4367 99226
5504 99630

For what concerns prime numbers, those which set a record are:
2 379
71 991
534 9901
1972 41149
2023 42643
4419 99901
where 379^2= 14+364+1
and the first decompositions for 991 are:
991^2 => 982+0+8+1
991^3 => 973+2+4+2+2+7+1
991^4 => 9+6+44+830+90+5+6+1
991^5 => 9+5+5+8+0+27+427+459+51
991^6 => 9+4+7+20+0+5+1+806+1+23+74+41
991^7 => 9+3+8+6+7+5+7+1+339+86+86+30+403+1
991^8 => 9+3+0+2+2+7+6+3+1+9+7+8+0+9+81+27+2+94+721
991^9 => 9+2+1+8+5+5+5+8+3+2+9+0+2+9+5+2+4+4+1+49+0+6+851+1
991^10 => 9+1+3+5+5+8+8+8+3+0+4+0+6+8+2+5+8+6+95+1+72+689+4+40+1
and so on, up to 991^71.

Just for curiosity, I also searched for prime powers which can be decomposed in sums with only prime terms.
I found these:
1063^5  => 13+5+727+0227+2+41+5+43
1297^7  => 61+7+41+877+61+97+53+7+7+5+61+7+13
7489^5  => 2+3+5+569+5+5+3+0311+137+6449
28429^4 => 653+20023+7+53+7+05+7681
63443^7 => 41+3+7+0043+5+0013+733+19913+23+2+042643+7+3+07
87103^5 => 5+013+7+83+37+7+2+557+47+85597+5+743

***

Jan wrote:

Below I give the maxima for the exponents. A * indicates I stopped searching, i.e. it will (probably) continue..
However the maximum value of the exponent is finite since the minimum sum of a split is equal to the sum of the digits.

Ending digit 0:
1{0},  exponent infinity (trivial solution)
990, exponent 107

Ending digit 1:
91, exponent 10
{9}[2]{0}[1]1, exponent 534
In this case splits can be found with numbers with no leading zeroes. (Only tested this one).

Similar patterns:
{9}[3]{0}[2]1, exponent 300*
{9}[4]{0}[3]1, exponent 365*
{9}[5]{0}[4]1, exponent 160*
{9}[6]{0}[5]1, exponent 160*
{9}[7]{0}[6]1, exponent 169*

Ending digit 2:
{9}[3]082, exponent 301*
{9}[10]082, exponent 390*
{9}[14]082, exponent 352*

Ending digit 3:
{9}[2]703, exponent 320*
{9}[11]703, exponent 120*

Ending digit 4:
{9}[2]964, exponent 301*

Ending digit 5:
{9}[2]45, exponent 311*
{9}[7]45, exponent 515*
{9}[2]55, exponent 322*
{9}[6]55, exponent 600*

Ending digit 6:
{9}[3]036, exponent 313*

Ending digit 7:
{9}[3]297, exponent 208*

Ending digit 8:
{9}[1]918, exponent 318*
{9}[12]918, exponent 200*

Ending digit 9:
{9}[k] only solution exponent 2
{9}[2]09, exponent 363*
{9}[7]09, exponent 502*

Above patterns come in pairs. The tails of the patterns at 5,1-9,2-8,3-7,4-6 all add to a power of 10.
Others do exist, like 1702, exponent 114 and 7777, exponent 443.

***

Hakan wrote:

There are many better solutions in these form.
My smallest  prime solution is 991^i  , for i=2 to 71.
Sum of digits[991^72]=1045 >991 , So There isn't any solution for i=72.

I send my solution in additional file (p668.txt).

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