The number of bases is unbounded Theorem Given any K ∈ N and d � 2 , there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each 1 � i � K, n is a d-digit palindrome in base b i . Proof (sketch). m ∈ N , τ ( m ) � 2 K + 1; a ′ 1 = 1 < · · · < a ′ K — the smallest K divisors of m ; a i = ( a ′ i ) d − 1 r n ❶ d − 1 ➀ d − 1 m ( d − 1) 2 , b i = n = ➎ d − 1 d − 1 − 1( ∈ N ) ➑ a i 2 ➤❶ d − 1 ➀ ❶ d − 1 ➀ ❶ d − 1 ➀ ❶ d − 1 ➀ ➳ n = a i , a i , . . . , a i , a i d − 1 d − 2 1 0 b i Bojan Baˇ si´ c On multipalindromic sequences 5/ 16
➡⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾➯ A further research direction Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
➡⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾➯ A further research direction Question Which palindromic sequences � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0, have the property that for any K ∈ N there exists a number that is a d -digit palindrome simultaneously in K different bases, with � c d − 1 , c d − 2 , . . . , c 0 � being its digit sequence in one of those bases? Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
➡⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾ ⑨ ❾➯ A further research direction Question Which palindromic sequences � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0, have the property that for any K ∈ N there exists a number that is a d -digit palindrome simultaneously in K different bases, with � c d − 1 , c d − 2 , . . . , c 0 � being its digit sequence in one of those bases? We shall refer to the sequences satisfying this condition as “very palindromic” sequences. Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
A further research direction Question Which palindromic sequences � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0, have the property that for any K ∈ N there exists a number that is a d -digit palindrome simultaneously in K different bases, with � c d − 1 , c d − 2 , . . . , c 0 � being its digit sequence in one of those bases? We shall refer to the sequences satisfying this condition as “very palindromic” sequences. ➡⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾➯ All the sequences , , , . . . , , , as d − 1 d − 2 d − 3 1 0 well as their multiples by a factor of form t d − 1 , are “very palindromic”. Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
A further research direction Question Which palindromic sequences � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0, have the property that for any K ∈ N there exists a number that is a d -digit palindrome simultaneously in K different bases, with � c d − 1 , c d − 2 , . . . , c 0 � being its digit sequence in one of those bases? We shall refer to the sequences satisfying this condition as “very palindromic” sequences. ➡⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾➯ All the sequences , , , . . . , , , as d − 1 d − 2 d − 3 1 0 well as their multiples by a factor of form t d − 1 , are “very palindromic”. These are the only ones known so far; we shall refer to them as “binomial sequences”. Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
A further research direction Question Which palindromic sequences � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0, have the property that for any K ∈ N there exists a number that is a d -digit palindrome simultaneously in K different bases, with � c d − 1 , c d − 2 , . . . , c 0 � being its digit sequence in one of those bases? We shall refer to the sequences satisfying this condition as “very palindromic” sequences. ➡⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾ ⑨ d − 1 ❾➯ All the sequences , , , . . . , , , as d − 1 d − 2 d − 3 1 0 well as their multiples by a factor of form t d − 1 , are “very palindromic”. These are the only ones known so far; we shall refer to them as “binomial sequences”. For d = 2, these are precisely all the palindromic sequences of length 2. Bojan Baˇ si´ c On multipalindromic sequences 6/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N , τ ( s ) � K Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
❳ ➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N , τ ( s ) � K , a 1 = 1 , a 2 , . . . , a K — divisors of s Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N , τ ( s ) � K , a 1 = 1 , a 2 , . . . , a K — divisors of s d − 1 ❳ c j m sj n = j =0 Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
➡ ➯ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N , τ ( s ) � K , a 1 = 1 , a 2 , . . . , a K — divisors of s d − 1 s ❳ c j m sj , b i = m n = ai j =0 Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
Easy comes first: palindromes of variable length Theorem Let d � 2 and a palindromic sequence � c d − 1 , c d − 2 , . . . , c 0 � , c d − 1 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a palindrome with at least d digits in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c d − 1 , c d − 2 , . . . , c 0 � b i 0 = n. Proof (sketch). m > max { c 0 , c 1 , . . . , c d − 1 } s ∈ N , τ ( s ) � K , a 1 = 1 , a 2 , . . . , a K — divisors of s d − 1 s ❳ c j m sj , b i = m n = ai j =0 ➡ ➯ n = c d − 1 , 0 , . . . , 0 , c d − 2 , 0 , . . . , 0 , c d − 3 , 0 , 0 , . . . , 0 , 0 , c 1 , 0 , . . . , 0 , c 0 b i ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ a i − 1 zeros a i − 1 zeros a i − 1 zeros Bojan Baˇ si´ c On multipalindromic sequences 7/ 16
Three digits — the main result Bojan Baˇ si´ c On multipalindromic sequences 8/ 16
Three digits — the main result Theorem Let a palindromic sequence � c 0 , c 1 , c 0 � , c 0 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a 3 -digit palindrome in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c 0 , c 1 , c 0 � b i 0 = n. Bojan Baˇ si´ c On multipalindromic sequences 8/ 16
Three digits — the main result Theorem Let a palindromic sequence � c 0 , c 1 , c 0 � , c 0 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a 3 -digit palindrome in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c 0 , c 1 , c 0 � b i 0 = n. Proof (sketch). Bojan Baˇ si´ c On multipalindromic sequences 8/ 16
Three digits — the main result Theorem Let a palindromic sequence � c 0 , c 1 , c 0 � , c 0 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a 3 -digit palindrome in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c 0 , c 1 , c 0 � b i 0 = n. Proof (sketch). First construction. Bojan Baˇ si´ c On multipalindromic sequences 8/ 16
Three digits — the main result Theorem Let a palindromic sequence � c 0 , c 1 , c 0 � , c 0 � = 0 , be given. Then for any K ∈ N there exists n ∈ N and a list of bases { b 1 , b 2 , . . . , b K } such that, for each i such that 1 � i � K, n is a 3 -digit palindrome in base b i , and that, for some i 0 such that 1 � i 0 � K, we have � c 0 , c 1 , c 0 � b i 0 = n. Proof (sketch). First construction. s — large enough, coprime to 1 , 2 , . . . , K − 1 , c 0 Bojan Baˇ si´ c On multipalindromic sequences 8/ 16
⑨ ❾ Three digits — the main result Proof (sketch). First construction. s — large enough, coprime to 1 , 2 , . . . , K − 1 , c 0 Bojan Baˇ si´ c On multipalindromic sequences 9/ 16
⑨ ❾ Three digits — the main result Proof (sketch). First construction. s — large enough, coprime to 1 , 2 , . . . , K − 1 , c 0 c 1 m ≡ − (mod s − c 0 ( K − 2)!) c 2 0 ( K − 2)! c 1 m ≡ − (mod s − 2 c 0 ( K − 2)!) 2 c 2 0 ( K − 2)! . . . c 1 m ≡ − 0 ( K − 2)! (mod s − ( K − 1) c 0 ( K − 2)!) ( K − 1) c 2 Bojan Baˇ si´ c On multipalindromic sequences 9/ 16
⑨ ❾ Three digits — the main result Proof (sketch). First construction. s — large enough, coprime to 1 , 2 , . . . , K − 1 , c 0 c 1 m ≡ − (mod s − c 0 ( K − 2)!) c 2 0 ( K − 2)! c 1 m ≡ − (mod s − 2 c 0 ( K − 2)!) 2 c 2 0 ( K − 2)! . . . c 1 m ≡ − 0 ( K − 2)! (mod s − ( K − 1) c 0 ( K − 2)!) ( K − 1) c 2 n = c 0 ( ms ) 2 + c 1 ms + c 0 Bojan Baˇ si´ c On multipalindromic sequences 9/ 16
Three digits — the main result Proof (sketch). First construction. s — large enough, coprime to 1 , 2 , . . . , K − 1 , c 0 c 1 m ≡ − (mod s − c 0 ( K − 2)!) c 2 0 ( K − 2)! c 1 m ≡ − (mod s − 2 c 0 ( K − 2)!) 2 c 2 0 ( K − 2)! . . . c 1 m ≡ − 0 ( K − 2)! (mod s − ( K − 1) c 0 ( K − 2)!) ( K − 1) c 2 ⑨ ❾ n = c 0 ( ms ) 2 + c 1 ms + c 0 , b i = m s − ( i − 1) c 0 ( K − 2)! Bojan Baˇ si´ c On multipalindromic sequences 9/ 16
➮ q Three digits — the main result Second construction. Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
➮ q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even pq � c 1 Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
q Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even pq � c 1 ➮ 1 < q c 0 +1 p < c 0 Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even pq � c 1 ➮ 1 < q c 0 +1 p < c 0 q g , h ∈ N , 1 < p g c 0 +1 q h < c 0 Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even pq � c 1 ➮ 1 < q c 0 +1 p < c 0 q g , h ∈ N , 1 < p g c 0 +1 q h < c 0 n = c 0 a 2 + c 1 a + c 0 for a = c 1 ( pq ) ( pq +1) M c 0 Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
Three digits — the main result Second construction. The case c 1 � = 0 . p , q ∈ N such that ln p ln q / ∈ Q c 0 | pq pq is even pq � c 1 ➮ 1 < q c 0 +1 p < c 0 q g , h ∈ N , 1 < p g c 0 +1 q h < c 0 n = c 0 a 2 + c 1 a + c 0 for a = c 1 ( pq ) ( pq +1) M c 0 n is a 3-digit palindrome in base a and p u i q v i ( pq + 1) i for ⌈ g +5 2 ⌉ � i � ⌈ g +1 2 ⌉ + K Bojan Baˇ si´ c On multipalindromic sequences 10/ 16
⑩ ❿ Three digits — the main result Second construction. Bojan Baˇ si´ c On multipalindromic sequences 11/ 16
⑩ ❿ Three digits — the main result Second construction. The case c 1 = 0 . Bojan Baˇ si´ c On multipalindromic sequences 11/ 16
Three digits — the main result Second construction. The case c 1 = 0 . 1 ⑩ c 0 + 1 ❿ p , q ∈ N such that 1 < q 2 K − 2 p < c 0 Bojan Baˇ si´ c On multipalindromic sequences 11/ 16
Three digits — the main result Second construction. The case c 1 = 0 . 1 ⑩ c 0 + 1 ❿ p , q ∈ N such that 1 < q 2 K − 2 p < c 0 n = c 0 a 2 + c 0 for a = ( pq ) K − 1 Bojan Baˇ si´ c On multipalindromic sequences 11/ 16
Three digits — the main result Second construction. The case c 1 = 0 . 1 ⑩ c 0 + 1 ❿ p , q ∈ N such that 1 < q 2 K − 2 p < c 0 n = c 0 a 2 + c 0 for a = ( pq ) K − 1 n is a 3-digit palindrome in base p K + i − 2 q K − i for 1 � i � K Bojan Baˇ si´ c On multipalindromic sequences 11/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives n = 3 726 430 975, Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives n = 3 726 430 975, n = � 1 , 5 , 1 � 61042 = � 1 , 11127 , 1 � 55734 = � 1 , 23473 , 1 � 50426 = � 1 , 37475 , 1 � 45118 . Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives n = 3 726 430 975, n = � 1 , 5 , 1 � 61042 = � 1 , 11127 , 1 � 55734 = � 1 , 23473 , 1 � 50426 = � 1 , 37475 , 1 � 45118 . The second construction gives Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
⑤ ④③ ⑥ ⑤ ④③ ⑥ ⑤ ④③ ⑥ Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives n = 3 726 430 975, n = � 1 , 5 , 1 � 61042 = � 1 , 11127 , 1 � 55734 = � 1 , 23473 , 1 � 50426 = � 1 , 37475 , 1 � 45118 . The second construction gives n = 79441 . . . 06401 , ⑤ ④③ ⑥ 10 418 005 digits Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
Three digits — examples � 1 , 5 , 1 � , K = 4 The first construction gives n = 3 726 430 975, n = � 1 , 5 , 1 � 61042 = � 1 , 11127 , 1 � 55734 = � 1 , 23473 , 1 � 50426 = � 1 , 37475 , 1 � 45118 . The second construction gives n = 79441 . . . 06401 , ⑤ ④③ ⑥ 10 418 005 digits n = � 1 , 5 , 1 � 2 9 653 618 · 3 4 826 809 · 5 = � 1 , 19906 . . . 06864 , 1 � 2 9 653 614 · 3 4 826 801 · 13 5 ⑤ ④③ ⑥ 5 209 003 digits = � 1 , 15179 . . . 59936 , 1 � 2 9 653 612 · 3 4 826 800 · 13 6 ⑤ ④③ ⑥ 5 209 003 digits = � 1 , 10550 . . . 83264 , 1 � 2 9 653 610 · 3 4 826 799 · 13 7 . ⑤ ④③ ⑥ 5 209 003 digits Bojan Baˇ si´ c On multipalindromic sequences 12/ 16
Three digits — examples Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives n = 375 223 562 302 052, Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives n = 375 223 562 302 052, n = � 2 , 0 , 2 � 13 697 145 = � 2 , 3 374 800 , 2 � 12 879 405 = � 2 , 6 985 440 , 2 � 12 061 665 = � 2 , 10 883 376 , 2 � 11 243 925 . Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives n = 375 223 562 302 052, n = � 2 , 0 , 2 � 13 697 145 = � 2 , 3 374 800 , 2 � 12 879 405 = � 2 , 6 985 440 , 2 � 12 061 665 = � 2 , 10 883 376 , 2 � 11 243 925 . The second construction gives Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives n = 375 223 562 302 052, n = � 2 , 0 , 2 � 13 697 145 = � 2 , 3 374 800 , 2 � 12 879 405 = � 2 , 6 985 440 , 2 � 12 061 665 = � 2 , 10 883 376 , 2 � 11 243 925 . The second construction gives n = 382 205 952 000 002, Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — examples � 2 , 0 , 2 � , K = 4 The first construction gives n = 375 223 562 302 052, n = � 2 , 0 , 2 � 13 697 145 = � 2 , 3 374 800 , 2 � 12 879 405 = � 2 , 6 985 440 , 2 � 12 061 665 = � 2 , 10 883 376 , 2 � 11 243 925 . The second construction gives n = 382 205 952 000 002, n = � 2 , 0 , 2 � 13 824 000 = � 2 , 3 571 200 , 2 � 12 960 000 = � 2 , 7 157 280 , 2 � 12 150 000 = � 2 , 10 773 182 , 2 � 11 390 625 . Bojan Baˇ si´ c On multipalindromic sequences 13/ 16
Three digits — comparison of the two constructions Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
Three digits — comparison of the two constructions The second construction seems much “worse” at the first glance Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
Three digits — comparison of the two constructions The second construction seems much “worse” at the first glance, but it is not necessarily so: Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
Three digits — comparison of the two constructions The second construction seems much “worse” at the first glance, but it is not necessarily so: For c 1 = 0 the second construction produces much smaller values on n than the first one as K becomes larger (for � 2 , 0 , 2 � and K = 20, we get a 151-digit number vs. a 724-digit number; for K = 100 we get a 1066-digit number vs. 31394-digit number). Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
Three digits — comparison of the two constructions The second construction seems much “worse” at the first glance, but it is not necessarily so: For c 1 = 0 the second construction produces much smaller values on n than the first one as K becomes larger (for � 2 , 0 , 2 � and K = 20, we get a 151-digit number vs. a 724-digit number; for K = 100 we get a 1066-digit number vs. 31394-digit number). The core of the second construction seems to provide some space for optimization in order to get a smaller number a (and thus a smaller number n ). Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
Three digits — comparison of the two constructions The second construction seems much “worse” at the first glance, but it is not necessarily so: For c 1 = 0 the second construction produces much smaller values on n than the first one as K becomes larger (for � 2 , 0 , 2 � and K = 20, we get a 151-digit number vs. a 724-digit number; for K = 100 we get a 1066-digit number vs. 31394-digit number). The core of the second construction seems to provide some space for optimization in order to get a smaller number a (and thus a smaller number n ). There are some arguments that suggest that for d > 3 the numbers we are looking for become much rarer; thus, it is not at all impossible that a construction that produces large values in the case d = 3 can be adapted to be of some use also for d > 3, while the one that produces small values in the case d = 3 actually only picks some exceptions whose existence essentially relies on the assumption d = 3. Bojan Baˇ si´ c On multipalindromic sequences 14/ 16
❳ ❳ ✒✖ ✗ ✓ ✒✖ ✗ ✓ More digits Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
❳ ❳ ✒✖ ✗ ✓ ✒✖ ✗ ✓ More digits What is known for d > 3? Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
❳ ❳ ✒✖ ✗ ✓ ✒✖ ✗ ✓ More digits What is known for d > 3? Almost nothing. Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
❳ ❳ ✒✖ ✗ ✓ ✒✖ ✗ ✓ More digits What is known for d > 3? Almost nothing. We present some heuristic arguments. For the sake of simplicity, we consider the sequence � 1 , 0 , 0 , . . . , 0 , 1 � . Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
✒✖ ✗ ✓ ✒✖ ✗ ✓ More digits What is known for d > 3? Almost nothing. We present some heuristic arguments. For the sake of simplicity, we consider the sequence � 1 , 0 , 0 , . . . , 0 , 1 � . The number of integers that are written as � 1 , 0 , 0 , . . . , 0 , 1 � a , for a � A , and that are palindromes with the same number of digits also in some other base, could be heuristically bounded above by A − 1 A − 1 1 1 ❳ ❳ − 2 ⌋− 1 . b ⌊ d d b ⌊ d 2 ⌋− d − 1 b =2 b =2 Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
More digits What is known for d > 3? Almost nothing. We present some heuristic arguments. For the sake of simplicity, we consider the sequence � 1 , 0 , 0 , . . . , 0 , 1 � . The number of integers that are written as � 1 , 0 , 0 , . . . , 0 , 1 � a , for a � A , and that are palindromes with the same number of digits also in some other base, could be heuristically bounded above by A − 1 A − 1 1 1 ❳ ❳ − 2 ⌋− 1 . b ⌊ d d b ⌊ d 2 ⌋− d − 1 b =2 b =2 If d � 6, then for A → ∞ the above value converges to ✒✖ d ✗ ✓ ✒✖ d ✗ ✓ d ζ − − ζ − 1 2 d − 1 2 Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
More digits What is known for d > 3? Almost nothing. We present some heuristic arguments. For the sake of simplicity, we consider the sequence � 1 , 0 , 0 , . . . , 0 , 1 � . The number of integers that are written as � 1 , 0 , 0 , . . . , 0 , 1 � a , for a � A , and that are palindromes with the same number of digits also in some other base, could be heuristically bounded above by A − 1 A − 1 1 1 ❳ ❳ − 2 ⌋− 1 . b ⌊ d d b ⌊ d 2 ⌋− d − 1 b =2 b =2 If d � 6, then for A → ∞ the above value converges to ✒✖ d ✗ ✓ ✒✖ d ✗ ✓ d ζ − − ζ − 1 , 2 d − 1 2 which is finite! Bojan Baˇ si´ c On multipalindromic sequences 15/ 16
A selection of open problems Bojan Baˇ si´ c On multipalindromic sequences 16/ 16
A selection of open problems Characterize all the “very palindromic” sequences for d > 3. Bojan Baˇ si´ c On multipalindromic sequences 16/ 16
A selection of open problems Characterize all the “very palindromic” sequences for d > 3. If this is too much to ask for, the following questions might be easier: Bojan Baˇ si´ c On multipalindromic sequences 16/ 16
A selection of open problems Characterize all the “very palindromic” sequences for d > 3. If this is too much to ask for, the following questions might be easier: Are sequences � 1 , 1 , . . . , 1 � and � 1 , 0 , 0 , . . . , 0 , 1 � “very palindromic” (for any d , or for each d )? Bojan Baˇ si´ c On multipalindromic sequences 16/ 16
A selection of open problems Characterize all the “very palindromic” sequences for d > 3. If this is too much to ask for, the following questions might be easier: Are sequences � 1 , 1 , . . . , 1 � and � 1 , 0 , 0 , . . . , 0 , 1 � “very palindromic” (for any d , or for each d )? Provide at least a single example of a “very palindromic” sequence, other than the “binomial sequences” (for d > 3), or prove that there are not any. Bojan Baˇ si´ c On multipalindromic sequences 16/ 16
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