[ *October 9, 2001* ]

An ODD_DDO puzzle question

1 3 5 7 9

This WONplate is totally dedicated to integers

composed of only ODD digits and whose prime factors are iDem DitO !

Is this a worthy challenge for you ?

Sequence A062016 will give you a good starting point

for the semiprime case (with exactly two prime factors).

Bear in mind that more than two prime factors are allowed.

Note that I am looking for ever larger integers/prime factors

that are sufficiently random. I don't want to highlight the obvious ones

that could be found too easily by reverse engineering like for instance

3 * Repunit(n).

or that could be derived from an infinite pattern like

(9)_{n}5 = 5 * 1(9)_{n}

Lastly if the digitlength of the factors are > 1 and near equal to each

other it certainly will enhance the beauty of the all-odd number found !

__Solutions__

Two prime factors

Integer ?_{all_odd} = f1 * f2

(possible if f1 and f2 > 7 ?)

Three prime factors

7335737 = 191 * 193 * 199

15995731573 = 1153 * 1933 * 7177

Four prime factors

111333 = 3 * 17 * 37 * 59

Five prime factors

7977553773 = 3 * 7 * 17 * 113 * 197753

Etc.

[ *June 5, 2002* ]

Jean Claude Rosa (email) solved the small question for 2 factors.

The answer is no. Every prime factor larger than 7 and 'all odd'

always ends in one of the following 20 numbers :

11, 13, 17, 19, 31, 33, 37, 39, 51, 53,

57, 59, 71, 73, 77, 79, 91, 93, 97 or 99.

If we multiply two by two any pair of these 20 numbers

we get the result that

The digit at the tens position is always even.

So for the first case with two prime factors there exist no solution

for factors larger than 7 of course.

Thus the products of two prime factors 'all odd' ends in

01, 03, 07, 09, 21, 23, 27, 29, 41, 43,

47, 49, 61, 63, 67, 69, 81, 83, 87 or 89.

If we next multiply two by two any pair of these 20 numbers

we still get the same result

The digit at the tens position is always even.

This means there is also no solution for the cases

with 4, 6, 8, ... prime factors 'all odd' and larger than 7.

Additional Question from Jean Claude Rosa [ *June 10, 2002* ]

" Is it possible to find a palindromic number 'all odd' as a result

of the product of 3, 5, ... prime factors 'all odd' ? "

[ *July 15, 2002* ]

Jean Claude Rosa calculated all the products of 3, 5 and 7

prime factors 'all odd' taken amongst 10 consecutive 'all odd'

prime numbers from 3 up to 3,999,999,979.

Here are the largest 'all odd' composites that he found

With 3 prime factors

391133599733953317333733 =

73131517 * 73131599 * 73133351

With 5 prime factors

775793373119 =

191 * 193 * 197 * 317 * 337

With 7 prime factors

3353535515 =

5 * 11 * 17 * 31 * 37 * 53 * 59

He also found some cute little 'all odd' palindromes !

**595** = 5 * 7 * 17

**3553** = 11 * 17 * 19

**19591** = 11 * 13 * 137

" J'espère avoir un peu de temps pour continuer

à rechercher des palindromes plus grands... "

[ *July 16, 2002* ]

This afternoon I (J. C. Rosa) discovered larger 'all odd'

palindromes, which of course made me very happy...

Here they are

With 3 prime factors

**9979531359799** = 19 * 71171 * 7379951

With 4 prime factors

**9935335335399** = 3 * 191 * 5171 * 3353153

With 5 prime factors

**9997159517999** = 19 * 53 * 59 * 937 * 179579

With 6 prime factors

**9919959599199** = 3 * 13 * 17 * 59 * 317 * 799991

He also found various palindromic composites with

prime factors NOT all different

**19399999391** = 59 * 59 * 5573111

**19975357991** = 53 * 53 * 7111199

And finally this curious one !

**777555777** = **3** * 37 * 73 * **95959**

[ *July 21, 2002* ]

J. C. Rosa communicates that the two largest

15-digit 'all odd' palindromes consisting

of 3 or 5 distinct prime factors are

**993957131759399** = 977 * 13537 * 75153751

**995393999393599** = 13 * 19 * 53 * 193 * 393971573

[ *October 21, 2002* ]

Edwin Clark (email) (site) looked for 'all odd' palindromes

with 'all odd' & 'palindromic' prime factors using *Maple*.

9 = 3 * 3

33 = 3 * 11

393 = 3 * 131

939 = 3 * 313

33933 = 3 * 11311

39993 = 3 * 13331

55 = 5 * 11

77 = 7 * 11

99 = 3 * 3 * 11

1331 = 11 * 11 * 11

3993 = 3 * 11 * 11 * 11

3773 = 7 * 7 * 7 * 11

**17377371** = 3 * 3 * 11 * 191 * 919

5775 = 3 * 5 * 5 * 7 * 11

Clearly some of these generalize. For example when 1333...3331 is

a prime we get one of the form **3999...9993**. The largest prime of

this form I found was one with 95 digits. [ For this I used Maple's

'isprime' function to determine primality.] Unfortunately all of these

have at least one small prime factor.

I found these by putting in a list the 43 odd palindromic primes

less than 10^5 and found all products of 2, 3, 4 and 5 of these. Then

I checked the products for oddness and palindromicity.

__p.s.__ primes of the form 1333...3331 are PDP primes

Plateau and Depression Primes

and are since then more extensively researched.

The second largest known probable prime of the form

(12*10^n-21)/9 or 4*(10^n-1)/3-1 is 1(3)_{17955}1

thus creating an all odd palindrome of 17957 digits !