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Diffstat (limited to 'README.md')
| -rw-r--r-- | README.md | 16 |
1 files changed, 13 insertions, 3 deletions
@@ -30,7 +30,7 @@ $$ De Weger did a computer search up to ${n\choose k} < 10^{30}$ in 1995. Now that we have faster computers we can go higher — and we can say for certain that the only solutions -up to ${n\choose k}<10^{41}$ are the ones found by de Weger: namely, the ones in the infinite +up to ${n\choose k}<10^{42}$ are the ones found by de Weger: namely, the ones in the infinite family above, $$ {15 \choose 5} = {14 \choose 6} = 3003 = {78 \choose 2}$$ $$ {104 \choose 39} = {103 \choose 40} = 61218182743304701891431482520$$ @@ -38,10 +38,20 @@ And the “sporadic” solutions: $$ {153 \choose 2} ={19\choose 5} = 11628$$ $$ {221\choose 2}={17 \choose 8}=24310$$ -This program searches up to $10^X$, when given the argument $X$. +This program searches up to ${n\choose k} <10^X$, when given the arguments `entry-limit X`. The search works by putting all ${n\choose k}<10^X$ with $5\leq k\leq \frac n 2$ into an array and sorting it. Then we check for adjacent elements which are equal, and use a binary search to check whether elements in the array are ${n\choose 2},{n\choose 3},{n\choose 4}$. This finds all solutions with $l>4$ (since the solutions with $l\leq 4$ are known). -Unfortunately searching to $10^{41}$ with this method already requires 13 GB of memory. +We can make a slight optimization by just storing the entries mod $2^{64}$ in the array, +then only doing the full comparison on entries who agree mod $2^{64}$. +But still searching to $10^{42}$ with this method already requires 11 GB of memory. + +Using this modular trick we can also search up to the 60,000th row (use arguments `row-limit 60000`), +and (sadly) confirm that the only repeated entries are the ones listed above and the new entries in the infinite family, +$$ {713\choose 273} = {714\choose 272} \approx 3.5 \times 10^{204}$$ +$$ {4894\choose 1870} = {4895\choose 1869} \approx 4.6 \times 10^{1141}$$ +$$ {33551 \choose 12816} = {33552 \choose 12815} \approx 6.0 \times 10^{9687}$$ +Again we run into a memory bottleneck — searching this far required 14 GB of memory. + |