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The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... (sequence A007304 in the OEIS) The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes.
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719. [1] The primes at the end of the concatenation in the Smarandache–Wellin primes are
Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal. In base 10, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., 9 appears in the reptend the same number of times as each other digit. [1]: 166 (For such primes in base 10, see OEIS: A073761.)
An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a (p−1)/2 equals () modulo p, where () is the Jacobi symbol. An Euler probable prime which is composite is called an Euler–Jacobi pseudoprime to base a. The smallest Euler-Jacobi pseudoprime to base 2 is 561.
A prime p (where p ≠ 2, 5 when working in base 10) is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q. [8]
A research of minimal primes in bases 2 to 30; Minimal primes and unsolved families in bases 2 to 30; Minimal primes and unsolved families in bases 28 to 50; J. Shallit, Minimal primes, Journal of Recreational Mathematics, 30:2, pp. 113–117, 1999-2000. PRP records, search by form 8*13^n+183 (primes of the form 8{0}111 in base 13), n=32020
Toggle the table of contents. ... 3249 = 57 2, palindromic in base 7 (12321 7), ... 3638 – sum of first 43 primes, 599th sphenic number;
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