"Binary" number wanted

Question

I search a 100-digit number, containing only the digits 0 and 1 ("binary" number ) and being the product of two distinct 50-digit-primes. Again, I search a method more efficient than simply creating binary-random-numbers and factoring them. To avoid misunterstanding, the "binary" number should, of course, be interpreted as a decimal number.

Answer 1

A first (half)-result : $$\small 10000000000000000000000000000000000000000000010111\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000000100001001011110000000000000000000000000000000101111011100010111$$

'Half' result because the second prime has $51$ digits and not $50$ but with the advantage of the two primes factors written with $0$ and $1$ only !

The second promising prime gave a large family of 'binary' solutions (this list is far from complete) : $$\small 10000000000000000000000000000000000000101000000101\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000000110101000010110000000000000000000000001010010101010111100000101$$

$$\small10000000000000000000000000000000000100010000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000010101001000000110000000000000000000001000110001000110010100000001$$ $$\small 10000000000000000000000000000000000100100000000101\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000010110001000010110000000000000000000001001010010001110110100000101$$ $$\small 10000000000000000000000000000000001000000000000011\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000100100001000001110000000000000000000010000100000001110001100000011$$ $$\small 10000000000000000000000000000000001000000001010001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000000100100001101000110000000000000000000010000100010101111000101010001$$ $$\small 10000000000000000000000000000000010000000001000101\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000001000100001100010110000000000000000000100001000010011110010101000101$$ $$\small 10000000000000000000000000000000010000000001010001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000001000100001101000110000000000000000000100001000010110111000101010001$$

$$\small 10000000000000000000000000000000010100000000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000001010100001000000110000000000000000000101001010000010110000100000001$$

$$\small 10000000000000000000000000000000100100010000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000010010101001000000110000000000000000001001010110001100110010100000001$$

$$\small 10000000000000000000000000000001000100100000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000100010110001000000110000000000000000010001101010011000110100100000001$$

$$\small 10000000000000000000000000000001001000000000010001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000100100100001001000110000000000000000010010100100001101011000100010001$$

$$\small 10000000000000000000000000000001001000000000011011\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000100100100001001101110000000000000000010010100100001111111101100011011$$

$$\small 10000000000000000000000000000001001000000001000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000100100100001100000110000000000000000010010100100011001110000101000001$$

$$\small 10000000000000000000000000000001001000100000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000100100110001000000110000000000000000010010101100011001010100100000001$$

$$\small 10000000000000000000000000000001110100000000000101\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000000111010100001000010110000000000000000011101111010001111110010100000101$$

$$\small 10000000000000000000000000000010000000000000011001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000001000000100001001100110000000000000000100001000000010110011100100011001$$

$$\small 10000000000000000000000000000010100000000000000011\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000001010000100001000001110000000000000000101001010000010100110001100000011$$

$$\small 10000000000000000000000000000010100000010000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000001010000101001000000110000000000000000101001010100011100010010100000001$$

$$\small 10000000000000000000000000000011100000000000000111\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000001110000100001000011110000000000000000111001110000011101110011100000111$$

$$\small 10000000000000000000000000000011111000000000000001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000001111100100001000000110000000000000000111111111100011111010000100000001$$

$$\small 10000000000000000000000000000100100000000000001001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000000000000000000000010010000100001000100110000000000000001001010010000100110010100100001001$$

$$\cdots$$

$$\small 10000000001000000000000000000000000000000001010001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000100000000000000000000000000100001101010110100000001000000000000000000010100111000101010001$$

$$\small 10000000001000000000000000000000000000001000001001\\\small\times 100000000000000000000000000000000000010000100000001=\\ \small 1000000000100000000000000000000000000100101000110110100000001000000000000000010000110010101100001001$$ $$\cdots$$

And we could continue this game for a long time or examine the families obtained when the first factor has only a few sparse $1$ and so on.

The initial problem (the product of two $50$ digits primes) can't have 'all binary solutions' like these. From a probabilistic point of view solutions should be much more sparse (if they exist at all). Anyway I'll let my code run a little for the first prime greater than $3\,10^{49}$.