In Book IX of The Elements, Euclid gave a method for constructing perfect numbers, although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist. Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers. In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers. To this day, it is not known if any odd perfect numbers exist, although numbers up to 10^1500 have been checked without success, making the existence of odd perfect numbers appear unlikely. The following table summarizes the development of ever-higher bounds for the smallest possible odd perfect number. author | bound Kanold | 10^20 Tuckerman | 10^36 Hagis | 10^50 Brent and Cohen | 10^160 Brent et al. | 10^300 Ochem and Rao | 10^1500 Euler showed that an odd perfect number, if it exists, must be of the form N = p^(4λ + 1) Q^2, where p is a prime of the form 4n + 1 (Fermat' s 4n + 1 theorem; Burton 1989), a result similar to that derived by Frenicle in 1657. In other words, an odd perfect number must be of the form N = p^α q_1^(2β_1) ...q_r^(2β_r) for distinct odd primes p, q_1, ..., q_r with p congruent α congruent 1 (mod 4). Steuerwald subsequently proved that the β_is cannot all be 1. Touchard proved that an odd perfect number, if it exists, must be of the form 12k + 1 or 36k + 9. In 1896, Stuyvaert stated that an odd perfect number must be a sum of two squares. In 1887, Sylvester conjectured and in 1925, Gradshtein proved that any odd perfect number must have at least six distinct prime factors. Hagis showed that odd perfect numbers must have at least eight distinct prime factors, in which case, the number is divisible by 15. In 1888, Catalan proved that if an odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime aliquot factors, and this was extended to 27 by Norton. Norton showed that odd perfect numbers not divisible by 3 or 5, it must have at least 15 distinct prime factors. Neilsen, improving the bound of Hagis, showed that if an odd perfect number is not divisible by 3, it must have at least 12 distinct prime factors. Nielsen also showed that a general odd perfect number, if it exists, must have at least 9 distinct prime factors. More recently, Hare has shown that any odd perfect number must have 75 or more prime factors. Improving this bound requires the factorization of several large numbers (Hare), and attempts are currently underway to perform these factorizations using the elliptic curve factorization method at mersenneforum.org and OddPerfect.org. Ochem and Rao subsequently showed that any odd perfect number has at least 101 not necessarily distinct prime factors. For the largest prime factor of an odd perfect number, Iannucci (1999, 2000) and Jenkins have worked to find lower bounds. The largest three factors must be at least 100000007, 10007, and 101. Goto and Ohno verified that the largest factor must be at least 100000007 using an extension to the methods of Jenkins. Ochem and Rao subsequently showed that the largest component (i.e., divisor p^a with p prime) is greater than 10^62. For the smallest prime factor of an odd perfect number with all even powers lower than six, Yamada determined an upper bound of exp(4.97401×10^10) For any odd perfect number with r prime factors and 1<=i<=5, Kishore has established upper bounds for small factors of odd perfect numbers by showing that p_i<2^(2^(t - i))(i).

even perfect number | odd number | Ore's conjecture | perfect number