##
**On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-\(p\) digits.**
*(English)*
Zbl 1479.11028

A Carmichael number is a composite number \(m\) such that the congruence \(a^m\equiv 1 \pmod{m}\) holds for all integers \(a\) coprime to \(m\). This congruence holds for all prime \(m\) by Fermat’s little theorem. This paper complements two different characterizations of Carmichael numbers, due to Korselt and Carmichael. It draws new and interesting connections between Carmichael numbers and the function \(s_p(m)\), where \(s_p(m)\) denotes the sum of the digits of \(m\) expressed in base \(p\).

This paper defines {primary Carmichael numbers}, a subset of the Carmichael numbers given by those squarefree \(m\) such that \(s_p(m)= p\) for all prime divisors of \(m\). It also defines a subset of the integers containing the Carmichael numbers, given by those squarefree \(m\) such that \(s_p(m)\geq p\) for all prime divisors of \(m\).

This paper then offers a new characterization of the Carmichael numbers in terms of the sum of digits function, as squarefree \(m\) satisfying \[\{m: p|m \text{ implies } s_p(m)\geq p,\quad s_p(m) \equiv 1 \pmod{p-1}\},\] complementing the characterizations of Korselt and Carmichael. It then proves some results expressing the denominators of the Bernoulli polynomials in terms of \(s_p(m)\). It also provides another explicit characterization of the Carmichael numbers in terms of polygonal numbers and \(s_p(m)\). The proofs follow from from clever elementary number theoretic considerations.

This paper defines {primary Carmichael numbers}, a subset of the Carmichael numbers given by those squarefree \(m\) such that \(s_p(m)= p\) for all prime divisors of \(m\). It also defines a subset of the integers containing the Carmichael numbers, given by those squarefree \(m\) such that \(s_p(m)\geq p\) for all prime divisors of \(m\).

This paper then offers a new characterization of the Carmichael numbers in terms of the sum of digits function, as squarefree \(m\) satisfying \[\{m: p|m \text{ implies } s_p(m)\geq p,\quad s_p(m) \equiv 1 \pmod{p-1}\},\] complementing the characterizations of Korselt and Carmichael. It then proves some results expressing the denominators of the Bernoulli polynomials in terms of \(s_p(m)\). It also provides another explicit characterization of the Carmichael numbers in terms of polygonal numbers and \(s_p(m)\). The proofs follow from from clever elementary number theoretic considerations.

Reviewer: Tanay V. Wakhare (Cambridge)

### MSC:

11A51 | Factorization; primality |

11B68 | Bernoulli and Euler numbers and polynomials |

11A63 | Radix representation; digital problems |

### Keywords:

Carmichael numbers; Bernoulli numbers and polynomials; Knödel numbers; polygonal numbers; denominator; sum of base-\(p\) digits; \(p\)-adic valuation
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\textit{B. C. Kellner} and \textit{J. Sondow}, Integers 21, Paper A52, 21 p. (2021; Zbl 1479.11028)

### Online Encyclopedia of Integer Sequences:

Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1 are all primes.

a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = p*q*r, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.

Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).

Least primary Carmichael number (A324316) with n prime factors, or -1 if no such number exists.

a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.

Squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p.

Primary Carmichael numbers.

Number of primary Carmichael numbers (A324316) less than 10^n.

Number of terms in A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) less than 10^n.

Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also hexagonal numbers (A000384) with index equal to their largest prime factor.

Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.

Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists.

Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p.

Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

Squarefree polygonal numbers P(s,n) with s >= 3 and n >= 3.

Special polygonal numbers.

Rank of the n-th special polygonal number A324973(n).

Rank of the n-th Carmichael number.

Rank of the n-th primary Carmichael number.

Denominator(Bernoulli_{m-1}) / m, where m is the n-th Carmichael number.

a(n) = n*denominator(n*Bernoulli(n-1)) for n >= 1 and a(0) = 0.

a(n) = denominator(b_n(x)), where b_n(x) are the polynomials defined in A335947.

Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

Positive integers k such that the fifth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

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