# A new upper bound for odd perfect numbers of a special form

title={A new upper bound for odd perfect numbers of a special form},
journal={Colloquium Mathematicum},
year={2019}
}
We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.
2 Citations
Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form
. We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.
An exponential diophantine equation related to odd perfect numbers
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give

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We show that there is an effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.
AN UPPER BOUND FOR ODD PERFECT NUMBERS
If N is an odd perfect number with k distinct prime factors then we show that N < 2 k . If some of the small prime factors of N are known then this bound can be further improved.
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If ra is a multiply perfect number (σ(m) = tm for some integer ί), we ask if there is a prime p with m = pan, (pa, n) = 1, σ(n) = pα, and σ(pa) = tn. We prove that the only multiply perfect numbers
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It is proved here that an odd number of the form paS6, where s is square-free, p is a prime which does not divide s, and p and a are both congruent to 1 modulo 4, cannot be perfect. A positive
EXTENSIONS OF SOME RESULTS CONCERNING ODD PERFECT NUMBERS
for distinct odd primes p, ql9 ..., qt, with p = a = 1 (mod 4). (We shall always assume this form for the prime factor decomposition of N) . Many writers have found conditions which must be satisfied
Introduction to Number Theory
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This seems simple enough, but let’s play with this definition. The Pythagoreans, an ancient sect of mathematical mystics, said that a number is perfect if it equals the sum of its positive integral
"J."
however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)
Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl.
Sind p, qv . . ., qr die in der ungeraden Zahl n aufgehenden verschiedenen Primteiler, so soll gezeigt werden: (1) Satz, n = pqlqlql · · · qr ist nicht vollkommen. Für die Fälle a = l und «. = 5 ist
Sätze über Kreisteilungspolynome und ihre Andwendungen auf einige zahlentheoretische Probleme. I.
In diesem Abschnitt sollen einige Sätze über Kreisteilungspolynome, die an anderer Stelle bewiesen wurden), verallgemeinert und durch einige Größenabschätzungen ergänzt werden. Wir wollen uns dabei