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  2. List of Mersenne primes and perfect numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_Mersenne_primes...

    [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.

  3. Square number - Wikipedia

    en.wikipedia.org/wiki/Square_number

    Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.

  4. Perfect number - Wikipedia

    en.wikipedia.org/wiki/Perfect_number

    Illustration of the perfect number status of the number 6. In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number.

  5. Table of prime factors - Wikipedia

    en.wikipedia.org/wiki/Table_of_prime_factors

    The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (sequence A005843 in the OEIS). An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form a 2 for some a).

  6. Happy number - Wikipedia

    en.wikipedia.org/wiki/Happy_number

    The palindromic prime 10 150006 + 7 426 247 × 10 75 000 + 1 is a 10-happy prime with 150 007 digits because the many 0s do not contribute to the sum of squared digits, and 1 2 + 7 2 + 4 2 + 2 2 + 6 2 + 2 2 + 4 2 + 7 2 + 1 2 = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.

  7. List of numbers - Wikipedia, the free encyclopedia

    en.wikipedia.org/wiki/List_of_numbers

    A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.

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    Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!

  9. Word square - Wikipedia

    en.wikipedia.org/wiki/Word_square

    The only perfect 10-squares published in any language to date have been constructed in Latin and English, and perfect 11-squares have been created in Latin as well. [11] Perfect 9-squares have been constructed in French, [12] while perfect squares of at least order 8 have been constructed in Italian and Spanish. [13]