Problems & Puzzles:
Puzzles
Puzzle 32.- Find couples of numbers
like this (1033, 8) such that:
1033 = 8^1+8^0+8^3+8^3
(Proposed by Patrick De Geest,
9/12/98)
This means to find couples of
numbers (N, B) such that N = sum( B^di ); di’s are
the decimal digits of N. Some other examples found by
Patrick are the following:
| N |
B |
Comments
about N |
| 1 |
1 |
Trivial |
| 10 |
9 |
|
| 12 |
3 |
|
| 100 |
98 |
|
| 101 |
50 |
Palprime |
| 111 |
37 |
Palindrome |
| 1033 |
8 |
Prime |
| 2112 |
32 |
Palindrome |
| 4624 |
4 |
|
| 20102 |
100 |
Palindrome |
| 31301 |
25 |
|
| 595968 |
4 |
|
| 1053202 |
16 |
|
| 3909511 |
5 |
|
| 13177388 |
7 |
|
| 52135640 |
19 |
|
Can you find
other prime and/or palindrome numbers N and its
corresponding base B ?
(See also: http://www.geocities.com/CapeCanaveral/Launchpad/4057/)
Solution
Felice Russo (4/06/99) sent solutions (N,B),
several for N=prime, one for N=Palprime, and two more for N=Palindrome:
(N prime, B) =
(101111, 20222)
(1010203,100)
(11111101,1587300)
(101001001, 25250249)
(N palprime, B) =
(111010111,15858587)
(N palindrome,B) =
(111101101101111, 9258425091759)
(101010000010101, 16835000001682)
***
This is the Patrick De
Geest's comment about the search of Russo:
"...I
didn't noticed it that with restricting oneself to binary
digits '0' and '1' one could construct more solutions.
Nice, especially the new palprime 111010111 with base
15858587. Searching for cases of N * X + d. (111010111 =
15858587 x 7 + 2 --> 7 units and 2 zeros)...Felice's
approach clearly is more effective. Congratulations..."
***
According to the Patrick's comment,
thanks to Russo's work, we have now at hand a method to
find larger ans larger examples of this kind, eh...?
***
The 13/8/2001 Tiziano Mosconi found the least
case (N,B) where N is palprime and B is prime:
{11100000011111000000111, 1009090910101000000009}
After this he found other 6 cases.
***
Anurag Sahay sent the following table (Jan 2005):
| N |
B |
Comments about N |
| 1224103 |
33 |
|
| 102531321 |
40 |
|
| 2758053616 |
15 |
|
| 7413245658 |
17 |
|
| 1347536041 |
20 |
|
| 2524232305 |
66 |
|
| 4303553602 |
40 |
|
| 16250130152 |
50 |
|
| 25236435456 |
48 |
|
| 35751665247 |
29 |
|
| 405287637330 |
28 |
|
| 419370838921 |
18 |
PRIME |
| 835713473952 |
21 |
|
| 985992657240 |
19 |
|
| 1035263675371 |
47 |
PRIME |
| 5063106413637 |
59 |
|
| 3606012949057 |
23 |
|
| 5398293152472 |
24 |
|
| 6821803782221 |
35 |
|
| 7560550222541 |
69 |
PRIME |
| 67850843167550 |
49 |
|
| 60383799081932 |
30 |
|
| 15554976231978 |
27 |
|
| 27348457391264 |
31 |
|
| 14375256503038 |
44 |
|
| 203200576992435 |
36 |
|
| 162455874783801 |
52 |
|
| 407374116975631 |
42 |
|
| 662953283249375 |
41 |
|
| 1382570662371937 |
48 |
|
| 3427807121982740 |
53 |
|
| 17918292835887535 |
59 |
|
| 27303134872679943 |
62 |
|
| 36609478778945537 |
64 |
PRIME |
| 42075899760411857 |
65 |
|
***
Later, on September 2005, Anurag Sahay sent
more examples:
(6035143058499257267, 113)
(680167723454978641921, 191)
(1470730546922115245998, 199)
(2459025789447836964220840587, 978)
(5144388812832249823657004916963778, 4928)
(36083215428634110977759 PRIME, 297)
(4034921884375455037142415526879 PRIME,2329)
***
On Set. 20, 06 Fred Schneider wrote:
I revisited this puzzle: I found one additional
solution: N=2210210, B=858 and found that one of the posted solutions
was incorrect: N=5063106413637, B=59 is not a solution because the
"power-digit-sum" = 2615920743659, not N itself. All the others check
out.
P.S. I'm very curious about Mr. Sahay's algorithm for
finding these solutions. It's very impressive
***
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