Calculation of special natural numbers

Question

As interested in factorization of integers, I had the idea to define the following natural numbers :

z(n) := [$\int_n^{n+1} x^x dx$]

My questions : 1) PARI can easily calculate z(n) numerically, but for large n it takes quite a long time. Is there an efficient method to calculate z(n) ? (The integral needs to be calculated with precision about 0.5) 2) The largest prime I found is z(275). Can anyone find bigger ones ? 3) z(79) is difficult to factor. I found the 34-digit prime-factor 1188 61814 33453 90012 05125 30694 68019. Can anyone factor the remaining cofactor with 119 digits ?

Answer 1

To be clear: do you have these values for $z(n)$?

$$ \begin{align} z(9) = & 2934896279 \\ z(10) = & 81607544630 \\ \end{align} $$

$$ \begin{align} z(25) = & 1427759269598032850533133531485164681 & \text{(37 digits)}\\ z(26) = & 101983795940249008487272274613469251169 & \text{(39 digits)}\\ \end{align} $$

$$ \begin{align} z(79) = & 3269053961273978708012560763055091753206667067617118000189124825327 \\ & 0570988621006960668913100798229081854831546404994677634140270246629 \\ & 445048206444354751_c \\ & \text{(152 digits)} \\ \\ = & 1188618143345390012051253069468019_p \times \\ \\ & 2750297881263329571824046756719154740522462783913046919066948130351 \\ & 3561309565945466309498365930814024227177872638493829_c \end{align} $$

(since the above seems to match your factors and the following has been proven prime by APRCL, it would be very surprising for these to be different values)

$$ \begin{align} z(275) = & 7404191871659104208983292411749579036693797273275810189154445217936 \\ & 8467539785423380425089304114782854116153515916169989356742884450414 \\ & 4559801172499309126587825244523783663393812553680157021618805963694 \\ & 7798501699634139078133395690188856029933501844025752694479861667486 \\ & 1660251950422497773626507007074206879521555579938034101830345982591 \\ & 1344230636448033427715283474786261266344944699639988373912291606495 \\ & 8280143851113390552378895598210161435974719562757471009957868468703 \\ & 1477834800616139218451950443738032002169647675171121492023021793178 \\ & 0500040547536539078580164652020723213547855727859376965297953795958 \\ & 2303663666173993184294926653400974771731334504129910369425994666844 \\ & 801_p \\ & \text{(673 digits)} \end{align} $$

Answer 2

Currently, the best suggestion I can make is to calculate $z(n)$ with a precision of at least $2 \lceil (n + 1) \log_k{(n + 1)} \rceil$, with PARI/GP preferring to use $k = 10$. For $n = 275$ that means using default(realprecision, 1348), which allows intnum(x = 275, 276, x^x) to calculate, after considerable time, an integer part of 2236 bits and a fractional part of 2244 bits. The fractional bits appear to be uniformly distributed (slightly biased toward $0$). The integer bits appear to be non-uniformly distributed (biased toward $1$).

It seems reasonable to expect the upper half of the significant digits to be correct, but that might not be justified.

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$z(79)$ is tough to factor. It seems that the numbers adjacent to its factors tend to involve very large primes, so there are no conveniently small subgroups to work in. Since $x^x$ behaves so wildly, $z(n)$ can probably be expected to produce a lot of "unstructured" composites.