Problems & Puzzles: Puzzles

Contributions came from Emmanuel Vantieghem, Oscar Volpatti

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Emmanuel wrote on Set 8, 2019

These are the solutions I found around 3 March 2019 and that I sent to Tomasz Ordowski :
46526102797441*116315256993601
111709172816656801*279272932041642001
177635683940025046467781066894531*710542735760100185871124267578121   (*)
2141993519227*34271896307617    (*)
A bit alter I found also :
291271*2621431    (*)
45511111*409599991   (*)
45953231799732552282931*229766158998662761414651
At the beginning of this week I found also :
57267349137931*1660753124999971
42376302249137634697112108001725217391*139236421675737942576225497719954285711
432489788904264079990671641791044776119402985074626865671641791*9658938618861897786458333333333333333333333333333333333333333311

Then I also discovered that some of my solutions figure at the OEIS (marked with (*)) in  A306479 :
squarefree composite numbers  m  such that  RAD(p - 1) = RAD(m - 1)  for every prime  p  dividing  m.
That sequence has many semiprimes.  But it is easy to show every phi-radical semiprime is in  A306479.
This follows immediately from the identity :
p*q - 1 = (p - 1)*(q - 1) + (p - 1) + (q - 1)     (1)
since a prime divisor of  p*q - 1  and  p - 1 (resp. q - 1)  must be in  q - 1 (resp. p - 1).

Here is how I found my solutions.
Suppose in  (1)  we have  Rad(p*q - 1) =Rad(p - 1) = RAD(q - 1) = R.
We distinguish three kind of primes in R.
kind 1 : primes  a  that occur in  p - 1  with same exponent as in  q - 1,
kind 2 : primes  b  that occur in  p - 1  with a strictly bigger exponent than in  q - 1,
kind 3 : primes  c  that occur in  p - 1  with a strictly smaller exponent than in  q - 1.
Then, after canceling  d  in both members of  (1)  there will remain an expression of the form :
L = A*M*N + B + C   (2).
where :
A  is the product of the factor of  p - 1 that consists of the primes  a  with the similar factor of  q - 1.
B  has only prime factors  b
C  has only prime factors  c
M  has only prime factors  b  with exponents at least one unit higher than in B
N  has only prime factors  c  with exponents at least one unit higher than in C
L  has only prime factors  a.
Of course one will have  B, C  coprime  and  rad(L)  divides  B + C.
In order to return to (1) we must multiply both members of (2) with the factor  d  which is the unique number such that
d*A*M*N = d*B * d*C,  i. e. : d = A*M*N / (B*C)  and which is automatically an integer.

Of course, we shall have  p - 1 = d*B  and  q - 1 = d*C  but wether of not  p  and  q  are prime is not at all sure.

A very simple example of an equation like (2) is :
3^k = 2^2 * 3X + 1 + 2
where you may take any odd  k > 1.
Here, X  is always of the form  (3^(k-1) - 1) / 4 (it will have 'foreing' prime factors : the ones that disappeared after dividing  (1)  by  d),

and  d  will be  (3^(k-1) -1 ) /2.
The smallest  k  that gives prime  p  and  q  is  541 (that solution can be seen in one of the comment lines of  A306479)

Here is how I found my biggest solution.
I tried to have for  L  a number of the form  2^u * 5^v * 7^w  that satisfies :
2^u * 5^v * 7^w = 2*5*7*(3^2)*(67^2)*X    + 3 + 67
So, I constructed the set of all numbers  m  of that form such that   (m - 70)  is divisible by  3^2 * 67^2.

(I examined also other combinations for  B + C = 70  without obtaining a solution).

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Oscar wrote on Set 13, 2019

I was able to add seven more entries to the list of phi-radical semiprime numbers:

m6510      = 1525781251 (known);
m291270    = 763546828801;
m132990    = 6031047559681;
m3038070   = 184597450297471;
m252210    = 732785991945841;
m15170370  = 18641350656000001;
m870870    = 55212580317094201;
m139543170 = 9969815738350374661.

This list is complete up to 1e16, but I suspect that there are no further solutions below 1e19.

Useful properties.

Every phi-radical semiprime number m=pq satisfies the following equality, noticed by Emmanuel Vantighem:
Let the squarefree number r be the common radical of the numbers m-1 and phi(m);
if a prime number f divides r, then it divides m-1 AND at least one factor of the product (p-1)(q-1);
wlog, suppose that f divides p-1; then it divides the difference (m-1)-(p-1) = p(q-1);
but f is coprime with p, so it divides q-1 too.

We can summarize the constraints on m, p, q as follows:
1) m = pq, with p < q;
2) p is prime;
3) q is prime;
4) m = ru+1;
5) p = rx+1;
6) q = ry+1;
For a given radical r, infinitely many numbers m satisfy properties 4 and 7.
The numbers m in the range m < r^3 can be easily checked, because at most one pair (p,q) satisfies properties 1,5,6.
Property 1 can be restated in terms of u,x,y as u = rxy+x+y;
if m < r^3, then u < r^2 and xy < r;
if xy = r-1 then x and y are coprime with r, violating properties 8 and 9;
therefore xy <= r-2, and x+y <= x + (r-2)/x <= r-1;
compute the coefficients a = u%r = x+y, b = (u-a)/r = xy;
find x and y as the roots of the quadratic equation z^2 - az + b = 0;
if such roots are different positive integers, check properties 2,3,8,9.

Finally, the primes 2 and 3 necessarily divide r.
(For all the solutions found, 5 divides r too).

I'll submit the factorizations of m,p,q for the new solutions, sorted by increasing r.

Solution 1
r = 132990;
m = 6031047559681;
p = 1462891;
q = 4122691;
m-1 = 2^9 * 3^12 * 5 * 11   * 13 * 31;
p-1 = 2   * 3    * 5 * 11^2 * 13 * 31;
q-1 = 2   * 3    * 5 * 11   * 13 * 31^2;

Solution 2
r = 252210;
m = 732785991945841;
p = 6053041;
q = 121060801;
m-1 = 2^4 * 3^3 * 5   * 7^10 * 1201;
p-1 = 2^4 * 3^2 * 5   * 7    * 1201;
q-1 = 2^6 * 3^2 * 5^2 * 7    * 1201;

Solution 3
r = 291270;
m = 763546828801;
p = 291271;
q = 2621431;
m-1 = 2^20 * 3   * 5^2 * 7 * 19 * 73;
p-1 = 2    * 3   * 5   * 7 * 19 * 73;
q-1 = 2    * 3^3 * 5   * 7 * 19 * 73;

Solution 4
r = 870870;
m = 55212580317094201;
p = 121921801;
q = 452852401;
m-1 = 2^3 * 3^10 * 5^2 * 7   * 11^6 * 13   * 29;
p-1 = 2^3 * 3    * 5^2 * 7^2 * 11   * 13   * 29;
q-1 = 2^4 * 3    * 5^2 * 7   * 11   * 13^2 * 29;

Solution 5
r = 3038070;
m = 184597450297471;
p = 3038071;
q = 60761401;
m-1 = 2   * 3^12 * 5   * 7^4 * 17 * 23 * 37;
p-1 = 2   * 3    * 5   * 7   * 17 * 23 * 37;
q-1 = 2^3 * 3    * 5^2 * 7   * 17 * 23 * 37;

Solution 6
r = 15170370;
m = 18641350656000001;
p = 45511111;
q = 409599991;
m-1 = 2^18 * 3^2 * 5^6 * 37 * 79 * 173;
p-1 = 2    * 3^2 * 5   * 37 * 79 * 173;
q-1 = 2    * 3^4 * 5   * 37 * 79 * 173;

Solution 7
r = 139543170;
m = 9969815738350374661;
p = 279086341;
q = 35723051521:
m-1 = 2^2 * 3^6 * 5 * 13 * 43^6 * 53 * 157;
p-1 = 2^2 * 3   * 5 * 13 * 43   * 53 * 157;
q-1 = 2^9 * 3   * 5 * 13 * 43   * 53 * 157;

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