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
***
Qu Shun Liang wrote (Dec 23, 2008) many solutions. I will
show only the prime solutions:
419370838921 18 prime!
36609478778945537 64 prime!
139251164815885201 80 prime!
3302927389750318219 101 prime!
947288765894401687681 198 prime!
181959980982018167249 149 prime!
1738302526177719712897 212 prime!
8229212226515738544409 252 prime!
8261738498910420005081 252 prime!
3083035169354427026797 226 prime!
72374194476756069205463 321 prime!
36083215428634110977759 297 prime!
95819846877371720464703 331 prime!
72486576766263162199063 321 prime!
22597217327585752660759 282 prime!
116357702912903726409230779 675 prime!
289308896713546025759668637 771 prime! &
89423567665158371901674138556551281, prime (25
digits), B=7080.
***
On July 13, 2018, Carlos Rivera wrote:
One more solution, a record one:
P=1825005747294136345232425233479077480431664917077
(Prime, 49 digits)
B=203868
***
On August 30, 2018, Carlos Rivera wrote:
One more solution, the second largest found by me:
P=1000283647546038884549781448648602016196015257
(Prime, 46 digits)
B=92590
***
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