Problems & Puzzles: Puzzles

 Puzzle 762. Conjecture from Ribenboim's book Luis Rodríguez sent the following puzzle base on a  CONJECTURE IN RIBENBOIM ́S “THE NEW BOOK OF PRIME NUMBER, (pag. 345) Conjecture: “If p is an odd prime then there exists one or more bases a, 2<= a < p  a>=2 such that a^(p-1) – 1 is divisible by p^2” From the tables 45 and 46 , Luis extracted the values for p <=211. (Table 46 show only prime bases) Prime p = 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Base a = 10 7 18 3 19 38 28 28 14 439 18 51 19 53 521 53 Prime p = 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 Base a = 601 ? 11 619 31 99 ? 53 617 43 317 96 68 38 58 19 ? Prime p = 149 151 157 163 167 173 179 181 191 193 197 199 211 Base a = 313 78 233 65 ? 419 ? 78 ? ? 997 ? 331 REMARK. The primes 1093 and 3511 are the only known corresponding to base 2. Q1.- Do you know a solid argument in favor of the conjecture? Q2.- To complete the values of a, signaled with a red question mark (?). Q3.- For the primes 31, 53, 61, 73, 101, 107, 149, 157, 173, 197 and 211 to search for smaller bases. Q4.- Find the  next p value with base=2.

Contributions came from Fred Schneider, Jan van Delden, Emmanuel Vantieghem, Jahangeer Kholdi & Farideh Firoozbakht

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Fred wrote:

Q1:
The moduli form a cyclic group of at most length p^2.  So, you would always have a solution <= p^2
(The exception is 3 with 10 but that's because of the smallness of the exponent used: 2)

Q3:
Due to basic modular math, if  base "a" is a solution for prime p,  a + c*p^2 (where c is a positive integer) will also be a solution.

Q4) base 2 solutions are called Wiefreich primes.  http://oeis.org/A001220
From there, there is this finding: "There are no other terms below 1.45*10^17"

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Jan wrote:

Q1:

I)

Let's first focus on:

# {a|2<=a<p, a^(p-1)=1 mod p^2}

We know
# {b|2<=b<p, (b^p)^(p-1)=1 mod p^2}=p-2 because we have (b,p^2)=(b,p)=1 and phi(p^2)=p*(p-1).
This induces a set of a:
# {a|2<=a<p, a=b^p mod p^2, 2<=b<p} (*)
Moreover since we have (b+k*p)^p mod p^2 = b^p mod p^2 (p>=2) the following set generates the same solutions:
# {a|2<=a<p, a=b^p mod p^2, 0<=b<p^2}
[Note we could restrict the allowed values of b further with b mod p not in [0,1].]
So in order to find all solutions we could restrict our attention to (*).

The values of b^p mod p^2 with 2<=b<p are (nearly) uniformly distributed on [0..p^2].
Let's define k: a solution is contained in [2..p-1]. The probability for this event is 1/p.
So k is Bin(p-2,1/p) distributed, with expectation 1 and variance 1 (nearly).
For large p this can be approximated by the Poisson(1) distribution.
[Note, strictly speaking we could restrict to [0..p^2]\S, with S={k: k mod p=0 or 1}].

So we found an approximation of the number of solutions to (*) as a Poisson(1) distribution.
We now have P(k>0)=1-P(k<=0)=1-exp(-1)=0.63.

An investigation of the distribution of the number of solutions to (*) by using all odd primes<10000 gives:

Fig1

Which gives: P(k>0)=0.66, mean=1.048 and variance=1.13. I would say reasonable.
Note: we have found two exceptional values k=10 and k=11, belonging to a=2, which have a rather big impact on the variance.
Or to put it differently P(k>=10)=1-P(k<=9)=1.11*10^(-7)

In Ribenboim, page 345:

<quote>Kruyswijk showed in 1966 that there exists a constant C such that for every odd prime p:

# {a|2<=a<p,a^(p-1)=1 mod p^2)} < p^(1/2+C/ln(ln p)).

So, not too many bases are good for each prime p.<end quote>

Given my investigation, I would say the upperbound is (way) too optimistic.
[But Kruyswijk found a provable upperbound].

II)

How about the smallest solution a (given the same set of b to try):

minimum {a|a=b^p mod p^2, 2<=b<p}

For all odd primes p<=10000 we have:

Fig2

This is very well approximated by the exponential(1) distribution: P(a/p<x) = 1-exp(-x).
So P(a<p)=P(a/p<1)=1-exp(-1)=0.63 again.

Q2,Q3: Just loop over a. For instance p=3 gives a=8 as minimum root. (see attachment).
Q4:      such a prime is a Wieferich prime, all p<4*10^12 have been checked already. A small article regarding properties of the two known Wieferich primes:
http://library.uwinnipeg.ca/people/dobson/mathematics/Wieferich_primes.html

...

If you change the puzzle in this fashion you might consider adding another constraint on a, because 1 solution generates an infinite number of solutions:

If b is a solution, so is a=b mod p^2.

Since p^2+1 is a solution, we have an infinite amount of solutions (i.e. we could take b=1).

So if you decide to lift the bound p, you  might consider changing it to p^2, otherwise the conjecture is not that hard... On the other hand, we than have values for p for which there are no solutions, so in that case the conjecture is easily falsified.

That's why it was formulated as a question and not like a conjecture in Ribenboim, I think.

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Emmanuel wrote:

Here is my contribution to puzzle 762.

The conjecture is a well known theorem in number theory.
In fact, in the interval  [ 1, p^2 ]  there are exactly  p-1  bases  a.  This is due to the fact that there exist a primitive root  r  modulo  p^2  whence you can take
a === r^(k*p)  for  k = 1, 2, ... , p-1  (this can be found in  A134307  which is the sequence of primes  p  for which one of the  a  is in the interval [ 2, p ] ).

Here is the list of pairs  { p , smallest  a } :
{3, 8}   ***
{5, 7}
{7, 18}
{11, 3}
{13, 19}
{17, 38}
{19, 28}
{23, 28}
{29, 14}
{31, 115}   ***
{37, 18}
{41, 51}
{43, 19}
{47, 53}
{53, 338}   ***
{59, 53}
{61, 264}   ***
{67, 143}     /?
{71,11}
{73, 306}   ***
{79, 31}
{83, 99}
{89, 184}     /?
{97,53}
{101, 181}   ***
{103, 43}
{107, 164}   ***
{109, 96}
{113, 68}
{127, 38}
{131, 58}
{137, 19}
{139, 328}     /?
{149,313}
{151, 78}
{157, 226}   ***
{163, 65}
{167, 253}     /?
{173, 259}   ***
{179, 532}     /?
{181, 78}
{191, 176}     /?
{193, 276}     /?
{197, 143}   ***
{199, 174}     /?
{211, 165}   ***
(here, *** means : smaller than in Rodrigues' table
/?  means : replaces the question mark.

It is not difficult to obtain this list.
For bigger  p,  I made a 'champions list' for the value  a/p.  This is what I found :
p                   a                      a/p
3                   8                    2.6666
31                 115                 3.7097
53                 338                 6.3774
6491             45329              6.9834
14431           108682            7.5311
16657           136471            8.1930
29129           295176          10.1334
The next champion is > 45000.

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Jahangeer Kholdi & Farideh Firoozbakht wrote:

Q1 : For every prime or composite number n, there exists at least one base b, b >1 such that n^2
divides b^(n-1) -1.

Proof : Take b = n^2 + 1, obviously n^2 divides (n^2+1)^(n-1) - 1 = b^(n-1) - 1.

Q2 & Q3 : Let a(n) be smallest b, b > 1 such that n^2 divides b^(n-1) - 1, then {p, a(p)} , for primes p,
p < = 211  :

{2, 5}, {3, 8}, {5, 7}, {7, 18}, {11, 3}, {13, 19}, {17, 38}, {19, 28}, {23, 28}, {29, 14}, {31, 115}, {37, 18},
{41, 51}, {43, 19}, {47, 53}, {53, 338}, {59, 53}, {61, 264}, {67, 143}, {71, 11}, {73, 306}, {79, 31}, {83, 99},
{89, 184}, {97, 53}, {101, 181}, {103, 43}, {107, 164}, {109, 96}, {113, 68}, {127, 38}, {131,   58}, {137, 19},
{139, 328}, {149, 313}, {151, 78}, {157, 226}, {163, 65}, {167, 253}, {173, 259}, {179, 532}, {181, 78},
{191, 176}, {193, 276}, {197, 143}, {199, 174} & {211, 165}.

Also a(n) for n = 1, 2, ..., 70  :

2, 5, 8, 17, 7, 37, 18, 65, 80, 101, 3, 145, 19, 197, 26, 257, 38, 325, 28, 401, 197, 485, 28, 577, 182,
677, 728, 177, 14, 901, 115,1025, 485, 1157, 99, 1297, 18, 1445, 170, 1601, 51, 1765, 19, 1937, 82,
2117, 53, 2305, 1047, 2501, 577, 529, 338, 2917, 1451, 3137, 721, 3365, 53, 3601, 264, 3845, 244,
4097, 99, 1945, 143, 4625, 530 & 3301.

Q4 : Next such prime p, if it exists is greater than 4*10^7.

Let b(n) be smallest prime p such that p^2 divides n^(p-1) -1 and zero if there is no such prime p.

b(n) for n = 1, 2, ..., 33 :

2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131,1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3,
11, 3, 2, 7, 7, 5 & 2.

It is interesting that for every n, b(n^(n^m)) = b(n) where m >= 0.

Find b(n) for n = 34, 47, 66, 72, 86, 88 & 90.

***

Richard Chen wrote on April 4, 2021_

For a(p) for primes p, see http://oeis.org/A039678
For a(n), see http://oeis.org/A185103
For b(n), see http://oeis.org/A039951

And b(34) = 46145917691, b(66) = 89351671, b(86) = 68239, b(88) = 2535619637, b(90) = 6590291053, also b(103) = 24490789, b(106) = 79399, b(135) = 23147324413, b(139) = 1822333408543, b(142) = 143111, and b(47) > 1.329*10^14, b(72) > 2.7*10^13, if they exist.

Besides, a(n) exists for all n, since at least, base a = n^2+1 satisfies the condition of a(n)

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