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[7] [8] [9] It is widely believed, [10] but not proven, that no odd perfect numbers exist; numerous restrictive conditions have been proven, [10] including a lower bound of 10 1500. [11] The following is a list of all 52 currently known (as of January 2025) Mersenne primes and corresponding perfect numbers, along with their exponents p.
A perfect pangram contains every letter of the alphabet only once and can be considered an anagram of the alphabet. The only perfect pangrams of the English alphabet that are known use abbreviations or other non-dictionary words, such as "Blowzy night-frumps vex'd Jack Q." or "Mr. Jock, TV quiz PhD, bags few lynx."
For example, a word where every featured letter appears twice, like "Shanghaiings", might be called a pair isogram, [8] a second-order isogram, [2] or a 2-isogram. [3] A perfect pangram is an example of a heterogram, with the added restriction that it uses all the letters of the alphabet.
An example would be Businessman burst into tears (9 letters). The solution, stationer, is an anagram of into tears, the letters of which have burst out of their original arrangement to form the name of a type of businessman. Numerous other games and contests involve some element of anagram formation as a basic skill. Some examples:
For example, Poe's poem The Raven contains no Z, but there is no evidence that this was intentional. A pangrammatic lipogram is a text that uses every letter of the alphabet except one. For example, "The quick brown fox jumped over the lazy dog" omits the letter S, which the usual pangram includes by using the word jumps.
The following phrases come from a portable media player's seven-segment display. They give a good illustration of an application where a seven-segment display may be sufficient for displaying letters, since the relevant messages are neither critical nor in any significant risk of being misunderstood, much due to the limited number and rigid domain specificity of the messages.
In the majority of alternades, every second letter is used to make two smaller words, but in some cases, every third letter is used to make three smaller words. Theoretically, a very long word could use every fourth letter to make four smaller words; e.g., «partitioned» is an alternade for «pin», «ate», «rid», and «to».
From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G ♯ 3—13 half-steps down from A4 is replicated as a B ♭ 5—13 half-steps above. Just as the letters of a verbal palindrome are not reversed, so are the elements of a musical palindrome usually presented in the same form in both halves.