Sorry if this is a faq, I am italian.
The question is: which is the cheapest way to go from Gatwick airport to
Heatrow airport (and viceversa)?
Outbound is on 19th may, inbound is 23rd may.
Thank you.

Probably £23.60:

£16 Standard Open Return (Southern/South Central) Gatwick - London
Victoria
Plus
2x £3.80 Zones 1-6 Single

From Gatwick Airport, catch a train to London Victoria marked Southern
or South Central (NOT Gatwick Express). At Victoria go down to the
Westbound District Line platform. Catch a train going to Ealing
Broadway or Richmond (NOT Wimbledon, Putney Bridge, Parson's Green,
Kensington Olympia, or Circle). At Baron's Court catch a Piccadilly
Line train to Heathrow Airport (NOT Rayner's Lane or Uxbridge).

Returning, take the first Piccadilly Line train to Hammersmith, and
change to the first District Line train across the platform (if the
destination is showing as High St Kensington, get off at Earl's Court
and wait for a train on the same platform (1/2 - either face) going to
Tower Hill, Upminster, Plaistow, Barking, or Whitechapel (not Edgware
Rd or High St Kensington)). Stay on until Victoria. At Victoria go up
into the half of the station with platforms 9-19. Get the first train
Southern/South Central that calls at Gatwick Airport (these tend to
show Hastings, Eastbourne, Littlehampton, Bognor Regis, Portsmouth, or
Southampton as their destination).