  We report unsuccessful searches for pulsations from the neutron star RX
J1836.2+5925 identified with the EGRET source 3EG J1835+5918. A 24 hr
observation with the NRAO Green Bank Telescope at 820 MHz placed an upper limit
on flux density of 17 uJy for P > 10 ms, and gradually increasing limits for 1
< P < 10 ms. The equivalent luminosity is lower than that of any known pulsar
with the possible exception of the radio-quiet gamma-ray pulsar Geminga. A set
of observations with the Chandra X-ray Observatory HRC totaling 118 ks revealed
no pulsar with 1 ms < P < 10 s. The upper limit on its pulsed fraction is 35%
assuming a sinusoidal pulse shape. The position of RX J1836.2+5925 in Chandra
observations separated by 3 years is unchanged within errors, leading to an
upper limit on its proper motion of <0.14"/yr, or v < 530 km/s at d = 800 pc, a
maximum distance estimated from its thermal X-ray spectrum. With these null
results, the properties of 3EG J1835+5918 and its X-ray counterpart RX
J1836.2+5925 are consistent with a more distant or older version of Geminga, or
perhaps a recycled pulsar. Having nearly exhausted the capabilities of current
instrumentation at all wavelengths, it will likely fall to the Gamma-ray Large
Area Space Telescope to discover pulsations from 3EG J1835+5918.
