The palindromic length PL(v) of a finite word v is the minimal number of palindromes whose concatenation is equal to v. In 2013, Frid, Puzynina, and Zamboni conjectured that: If w is an infinite word and k is an integer such that PL(u) ≤ k for every factor u of w then w is ultimately periodic. Suppose that w is an infinite word and k is an integer such PL(u) ≤ k for every factor u of w. Let Ω(w, k) be the set of all factors u of w that have more than ~k√(k~(-1)|u|) palindromic prefixes. We show that Ω(w, k) is an infinite set and we show that for each positive integer j there are palindromes a, b and a word u ∈ Ω{w, k) such that (ab)~j is a factor of u and b is nonempty. Note that (ab)~j is a periodic word and (ab)~i a is a palindrome for each i ≤ j. These results justify the following question: What is the palindromic length of a concatenation of a suffix of b and a periodic word (ab)~j with "many" periodic palindromes? It is known that if u, v are nonempty words then |PL(uv) - PL(u)| ≤ PL(v). The main result of our article shows that if a, b are palindromes, b is nonempty, u is a nonempty suffix of b, |ab| is the minimal period of aba, and j is a positive integer with j ≥ 3PL(u) then PL(u(ab)~j) - PL(u) ≥ 0.
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机译：有限字V的回文长度Pl（v）是倾斜等于v的最小数量的回文。2013年，Frid，puzynina和zamboni猜明：如果w是无限的单词，k是一个整数每个因子U的PL（U）≤K然后w最终定期。假设W是无限的单词，并且k是W的整数u（u）≤k。让ω（w，k）是w的所有因素U的设置超过〜k√（k〜（-1）| U |）回文前缀。我们表明ω（w，k）是无限的集合，我们向每个正整数j显示出有polindromes a，b和一个字U∈ω{w，k），使得（ab）~j是一个因子u和b是非空的。请注意（ab）〜j是周期性的单词和（ab）〜i a a是每个i≤j的回文。这些结果证明了以下问题：B的后缀和周期词（AB）〜J的串联的级联的回音长度是什么？众所周知，如果你，v是非空的单词| pl（uv） - pl（u）| ≤PL（v）。我们文章的主要结果表明，如果a，b是palindromes，b是nonempty，你是b的非空的后缀，| ab |是ABA的最小时期，J是具有J≥3pl（u）的正整数，然后pl（u（ab）〜j） - pl（u）≥0。
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