I created the following number
$$135791113\cdots 2017201920212023$$ which emerges by writing down the odd numbers from $1$ to $2023$ in increasing order. Its full decimal expansion can be seen here. As factordb shows (and PARI/GP approves) , this number is composite , but it has no small prime factor.
I searched prime factors with PARI/GP and the online magma calculator and also with Alpertron. They all seem too slow, so I think I have to use yafu (or GMP ECM)
Unfortunately, I have only yafu versions that cannot handle long expressions (possibly because of my old computer / system) . Therefore my question :
Is there a short expression giving this number ?
It should contain only the fundamental operators +-./^ and quite small numbers. Using the structure I could compress the number to about the half , but it is still too long.
If wished, I can post this expression, but I do not think that it is actually helpful.
If someone has access to an efficient tool , prime factors of this number are appreciated.
You can write it as $$13579\cdot 10^{3488}+10^{3398}\sum_{k=0}^{44} (11+2k)10^{2(44-k)}+10^{2048}\sum_{k=0}^{449}(101+2k)10^{3(449-k)}+\sum_{k=0}^{511}(1001+2 k)10^{4(511-k)}$$
or equivalently
$$13579\cdot 10^{3488}+10^{3398}\frac{1091\cdot 10^{90}-10001}{9801}+10^{2048}\frac{100901\cdot 10^{1350}-1000001}{998001}+\frac{10009001\cdot 10^{2048}-20247977}{99980001}$$
but I doubt those expressions turn out to be useful to investigate the factors of the number.